Digit Spiral

Tout a commencé par un message à la liste SeqFans:

1) start to write on a grid the nonnegative integers so to form a spiral (one digit clockwise per unit-square);

2) make a pause for the last integer fitting perfectly in a [square of unit-squares] and leaving no “hole” (consider both SW et NE corners!) -- else proceed;

3) colour this last integer in yellow;

4) call Z the sequence of yellow terms.

Spiral Z side of square

0 -> 0 Sq = 1

0 1

3 2 -> 3 Sq = 2

6 7 8 -> 8 Sq = 3

5 0 1

4 3 2

6 7 8 9

5 0 1 1

4 3 2 0

2 1 1 1 -> 12 Sq = 4

1 5 1 6 1 7

4 6 7 8 9 1

1 5 0 1 1 8

3 4 3 2 0 1

1 2 1 1 1 9

2 2 1 2 0 2 -> 22 Sq = 6

2 6 2 7 2 8 2 9

5 1 5 1 6 1 7 3

2 4 6 7 8 9 1 0

4 1 5 0 1 1 8 3

2 3 4 3 2 0 1 1

3 1 2 1 1 1 9 3

2 2 2 1 2 0 2 2

6 3 5 3 4 3 3 3 -> 36 Sq = 8

4 1 4 2 4 3 4 4 4 5

0 2 6 2 7 2 8 2 9 4

4 5 1 5 1 6 1 7 3 6

9 2 4 6 7 8 9 1 0 4

3 4 1 5 0 1 1 8 3 7

8 2 3 4 3 2 0 1 1 4

3 3 1 2 1 1 1 9 3 8

7 2 2 2 1 2 0 2 2 4

3 6 3 5 3 4 3 3 3 9

4 5 3 5 2 5 1 5 0 5 -> 54 Sq = 10

6 0 6 1 6 2 6 3 6 4 6 5

9 4 1 4 2 4 3 4 4 4 5 6

5 0 2 6 2 7 2 8 2 9 4 6

8 4 5 1 5 1 6 1 7 3 6 6

5 9 2 4 6 7 8 9 1 0 4 7

7 3 4 1 5 0 1 1 8 3 7 6

5 8 2 3 4 3 2 0 1 1 4 8

6 3 3 1 2 1 1 1 9 3 8 6

5 7 2 2 2 1 2 0 2 2 4 9

5 3 6 3 5 3 4 3 3 3 9 7

5 4 5 3 5 2 5 1 5 0 5 0

6 7 5 7 4 7 3 7 2 7 1 7 -> 76 Sq = 12

8 3 8 4 8 5 8 6 8 7 8 8 8 9

2 6 0 6 1 6 2 6 3 6 4 6 5 9

8 9 4 1 4 2 4 3 4 4 4 5 6 0

1 5 0 2 6 2 7 2 8 2 9 4 6 9

8 8 4 5 1 5 1 6 1 7 3 6 6 1

0 5 9 2 4 6 7 8 9 1 0 4 7 9

8 7 3 4 1 5 0 1 1 8 3 7 6 2

9 5 8 2 3 4 3 2 0 1 1 4 8 9

7 6 3 3 1 2 1 1 1 9 3 8 6 3

8 5 7 2 2 2 1 2 0 2 2 4 9 9

7 5 3 6 3 5 3 4 3 3 3 9 7 4

7 5 4 5 3 5 2 5 1 5 0 5 0 9

7 6 7 5 7 4 7 3 7 2 7 1 7 5

1 0 1 0 0 1 9 9 8 9 7 9 6 9 -> 101 Sq = 14

6 1 0 7 1 0 8 1 0 9 1 1 0 1 1 1

0 8 3 8 4 8 5 8 6 8 7 8 8 8 9 1

1 2 6 0 6 1 6 2 6 3 6 4 6 5 9 1

5 8 9 4 1 4 2 4 3 4 4 4 5 6 0 2

0 1 5 0 2 6 2 7 2 8 2 9 4 6 9 1

1 8 8 4 5 1 5 1 6 1 7 3 6 6 1 1

4 0 5 9 2 4 6 7 8 9 1 0 4 7 9 3

0 8 7 3 4 1 5 0 1 1 8 3 7 6 2 1

1 9 5 8 2 3 4 3 2 0 1 1 4 8 9 1

3 7 6 3 3 1 2 1 1 1 9 3 8 6 3 4

0 8 5 7 2 2 2 1 2 0 2 2 4 9 9 1

1 7 5 3 6 3 5 3 4 3 3 3 9 7 4 1

2 7 5 4 5 3 5 2 5 1 5 0 5 0 9 5

0 7 6 7 5 7 4 7 3 7 2 7 1 7 5 1

1 1 0 1 0 0 1 9 9 8 9 7 9 6 9 1

1 2 1 0 2 1 9 1 1 8 1 1 7 1 1 6 -> 121 Sq = 16

2 7 1 2 8 1 2 9 1 3 0 1 3 1 1 3 2 -> 132 Sq = 17

1 6 1 0 7 1 0 8 1 0 9 1 1 0 1 1 1

6 0 8 3 8 4 8 5 8 6 8 7 8 8 8 9 1

2 1 2 6 0 6 1 6 2 6 3 6 4 6 5 9 1

1 5 8 9 4 1 4 2 4 3 4 4 4 5 6 0 2

5 0 1 5 0 2 6 2 7 2 8 2 9 4 6 9 1

2 1 8 8 4 5 1 5 1 6 1 7 3 6 6 1 1

1 4 0 5 9 2 4 6 7 8 9 1 0 4 7 9 3

4 0 8 7 3 4 1 5 0 1 1 8 3 7 6 2 1

2 1 9 5 8 2 3 4 3 2 0 1 1 4 8 9 1

1 3 7 6 3 3 1 2 1 1 1 9 3 8 6 3 4

3 0 8 5 7 2 2 2 1 2 0 2 2 4 9 9 1

2 1 7 5 3 6 3 5 3 4 3 3 3 9 7 4 1

1 2 7 5 4 5 3 5 2 5 1 5 0 5 0 9 5

2 0 7 6 7 5 7 4 7 3 7 2 7 1 7 5 1

2 1 1 0 1 0 0 1 9 9 8 9 7 9 6 9 1

1 1 2 1 0 2 1 9 1 1 8 1 1 7 1 1 6

... etc.

Could someone compute more terms? I guess we have for a start S = 0,3,8,12,22,36,54,76,101,121,132, ...

(hope this is not old hat)

Best,

É.

(yes, I know, I’m colorblind :-)

__________

Jean-Marc Falcoz a dégainé plus vite que son ombre et calculé tous les termes < 10 000 000 :

Z = 0, 3, 8, 12, 22, 36, 54, 76, 101, 121, 132, 156, 169, 197, 212, 244, 261, 297, 316, 356, 377, 421, 444, 492, 517, 569, 596, 652, 681, 741, 772, 836, 869, 937, 972, 10221, 10626, 11041, 11466, 11901, 12346, 12801, 13266, 13741, 14226, 14721, 15226, 15741, 16266, 16801, 17346, 17901, 18466, 19041, 19626, 20221, 20826, 21441, 22066, 22701, 23346, 24001, 24666, 25341, 26026, 26721, 27426, 28141, 28866, 29601, 30346, 31101, 31866, 32641, 33426, 34221, 35026, 35841, 36666, 37501, 38346, 39201, 40066, 40941, 41826, 42721, 43626, 44541, 45466, 46401, 47346, 48301, 49266, 50241, 51226, 52221, 53226, 54241, 55266, 56301, 57346, 58401, 59466, 60541, 61626, 62721, 63826, 64941, 66066, 67201, 68346, 69501, 70666, 71841, 73026, 74221, 75426, 76641, 77866, 79101, 80346, 81601, 82866, 84141, 85426, 86721, 88026, 89341, 90666, 92001, 93346, 94701, 96066, 97441, 98826, 100184, 101120, 101590, 102534, 103008, 103960, 104438, 105398, 105880, 106848, 107334, 108310, 108800, 109784, 110278, 111270, 111768, 112768, 113270, 114278, 114784, 115800, 116310, 117334, 117848, 118880, 119398, 120438, 120960, 122008, 122534, 123590, 124120, 125184, 125718, 126790, 127328, 128408, 128950, 130038, 130584, 131680, 132230, 133334, 133888, 135000, 135558, 136678, 137240, 138368, 138934, 140070, 140640, 141784, 142358, 143510, 144088, 145248, 145830, 146998, 147584, 148760, 149350, 150534, 151128, 152320, 152918, 154118, 154720, 155928, 156534, 157750, 158360, 159584, 160198, 161430, 162048, 163288, 163910, 165158, 165784, 167040, 167670, 168934, 169568, 170840, 171478, 172758, 173400, 174688, 175334, 176630, 177280, 178584, 179238, 180550, 181208, 182528, 183190, 184518, 185184, 186520, 187190, 188534, 189208, 190560, 191238, 192598, 193280, 194648, 195334, 196710, 197400, 198784, 199478, 200870, 201568, 202968, 203670, 205078, 205784, 207200, 207910, 209334, 210048, 211480, 212198, 213638, 214360, 215808, 216534, 217990, 218720, 220184, 220918, 222390, 223128, 224608, 225350, 226838, 227584, 229080, 229830, 231334, 232088, 233600, 234358, 235878, 236640, 238168, 238934, 240470, 241240, 242784, 243558, 245110, 245888, 247448, 248230, 249798, 250584, 252160, 252950, 254534, 255328, 256920, 257718, 259318, 260120, 261728, 262534, 264150, 264960, 266584, 267398, 269030, 269848, 271488, 272310, 273958, 274784, 276440, 277270, 278934, 279768, 281440, 282278, 283958, 284800, 286488, 287334, 289030, 289880, 291584, 292438, 294150, 295008, 296728, 297590, 299318, 300184, 301920, 302790, 304534, 305408, 307160, 308038, 309798, 310680, 312448, 313334, 315110, 316000, 317784, 318678, 320470, 321368, 323168, 324070, 325878, 326784, 328600, 329510, 331334, 332248, 334080, 334998, 336838, 337760, 339608, 340534, 342390, 343320, 345184, 346118, 347990, 348928, 350808, 351750, 353638, 354584, 356480, 357430, 359334, 360288, 362200, 363158, 365078, 366040, 367968, 368934, 370870, 371840, 373784, 374758, 376710, 377688, 379648, 380630, 382598, 383584, 385560, 386550, 388534, 389528, 391520, 392518, 394518, 395520, 397528, 398534, 400550, 401560, 403584, 404598, 406630, 407648, 409688, 410710, 412758, 413784, 415840, 416870, 418934, 419968, 422040, 423078, 425158, 426200, 428288, 429334, 431430, 432480, 434584, 435638, 437750, 438808, 440928, 441990, 444118, 445184, 447320, 448390, 450534, 451608, 453760, 454838, 456998, 458080, 460248, 461334, 463510, 464600, 466784, 467878, 470070, 471168, 473368, 474470, 476678, 477784, 480000, 481110, 483334, 484448, 486680, 487798, 490038, 491160, 493408, 494534, 496790, 497920, 500184, 501318, 503590, 504728, 507008, 508150, 510438, 511584, 513880, 515030, 517334, 518488, 520800, 521958, 524278, 525440, 527768, 528934, 531270, 532440, 534784, 535958, 538310, 539488, 541848, 543030, 545398, 546584, 548960, 550150, 552534, 553728, 556120, 557318, 559718, 560920, 563328, 564534, 566950, 568160, 570584, 571798, 574230, 575448, 577888, 579110, 581558, 582784, 585240, 586470, 588934, 590168, 592640, 593878, 596358, 597600, 600088, 601334, 603830, 605080, 607584, 608838, 611350, 612608, 615128, 616390, 618918, 620184, 622720, 623990, 626534, 627808, 630360, 631638, 634198, 635480, 638048, 639334, 641910, 643200, 645784, 647078, 649670, 650968, 653568, 654870, 657478, 658784, 661400, 662710, 665334, 666648, 669280, 670598, 673238, 674560, 677208, 678534, 681190, 682520, 685184, 686518, 689190, 690528, 693208, 694550, 697238, 698584, 701280, 702630, 705334, 706688, 709400, 710758, 713478, 714840, 717568, 718934, 721670, 723040, 725784, 727158, 729910, 731288, 734048, 735430, 738198, 739584, 742360, 743750, 746534, 747928, 750720, 752118, 754918, 756320, 759128, 760534, 763350, 764760, 767584, 768998, 771830, 773248, 776088, 777510, 780358, 781784, 784640, 786070, 788934, 790368, 793240, 794678, 797558, 799000, 801888, 803334, 806230, 807680, 810584, 812038, 814950, 816408, 819328, 820790, 823718, 825184, 828120, 829590, 832534, 834008, 836960, 838438, 841398, 842880, 845848, 847334, 850310, 851800, 854784, 856278, 859270, 860768, 863768, 865270, 868278, 869784, 872800, 874310, 877334, 878848, 881880, 883398, 886438, 887960, 891008, 892534, 895590, 897120, 900184, 901718, 904790, 906328, 909408, 910950, 914038, 915584, 918680, 920230, 923334, 924888, 928000, 929558, 932678, 934240, 937368, 938934, 942070, 943640, 946784, 948358, 951510, 953088, 956248, 957830, 960998, 962584, 965760, 967350, 970534, 972128, 975320, 976918, 980118, 981720, 984928, 986534, 989750, 991360, 994584, 996198, 999430, 1001592, 1006457, 1011336, 1016229, 1021136, 1026057, 1030992, 1035941, 1040904, 1045881, 1050872, 1055877, 1060896, 1065929, 1070976, 1076037, 1081112, 1086201, 1091304, 1096421, 1101552, 1106697, 1111856, 1117029, 1122216, 1127417, 1132632, 1137861, 1143104, 1148361, 1153632, 1158917, 1164216, 1169529, 1174856, 1180197, 1185552, 1190921, 1196304, 1201701, 1207112, 1212537, 1217976, 1223429, 1228896, 1234377, 1239872, 1245381, 1250904, 1256441, 1261992, 1267557, 1273136, 1278729, 1284336, 1289957, 1295592, 1301241, 1306904, 1312581, 1318272, 1323977, 1329696, 1335429, 1341176, 1346937, 1352712, 1358501, 1364304, 1370121, 1375952, 1381797, 1387656, 1393529, 1399416, 1405317, 1411232, 1417161, 1423104, 1429061, 1435032, 1441017, 1447016, 1453029, 1459056, 1465097, 1471152, 1477221, 1483304, 1489401, 1495512, 1501637, 1507776, 1513929, 1520096, 1526277, 1532472, 1538681, 1544904, 1551141, 1557392, 1563657, 1569936, 1576229, 1582536, 1588857, 1595192, 1601541, 1607904, 1614281, 1620672, 1627077, 1633496, 1639929, 1646376, 1652837, 1659312, 1665801, 1672304, 1678821, 1685352, 1691897, 1698456, 1705029, 1711616, 1718217, 1724832, 1731461, 1738104, 1744761, 1751432, 1758117, 1764816, 1771529, 1778256, 1784997, 1791752, 1798521, 1805304, 1812101, 1818912, 1825737, 1832576, 1839429, 1846296, 1853177, 1860072, 1866981, 1873904, 1880841, 1887792, 1894757, 1901736, 1908729, 1915736, 1922757, 1929792, 1936841, 1943904, 1950981, 1958072, 1965177, 1972296, 1979429, 1986576, 1993737, 2000912, 2008101, 2015304, 2022521, 2029752, 2036997, 2044256, 2051529, 2058816, 2066117, 2073432, 2080761, 2088104, 2095461, 2102832, 2110217, 2117616, 2125029, 2132456, 2139897, 2147352, 2154821, 2162304, 2169801, 2177312, 2184837, 2192376, 2199929, 2207496, 2215077, 2222672, 2230281, 2237904, 2245541, 2253192, 2260857, 2268536, 2276229, 2283936, 2291657, 2299392, 2307141, 2314904, 2322681, 2330472, 2338277, 2346096, 2353929, 2361776, 2369637, 2377512, 2385401, 2393304, 2401221, 2409152, 2417097, 2425056, 2433029, 2441016, 2449017, 2457032, 2465061, 2473104, 2481161, 2489232, 2497317, 2505416, 2513529, 2521656, 2529797, 2537952, 2546121, 2554304, 2562501, 2570712, 2578937, 2587176, 2595429, 2603696, 2611977, 2620272, 2628581, 2636904, 2645241, 2653592, 2661957, 2670336, 2678729, 2687136, 2695557, 2703992, 2712441, 2720904, 2729381, 2737872, 2746377, 2754896, 2763429, 2771976, 2780537, 2789112, 2797701, 2806304, 2814921, 2823552, 2832197, 2840856, 2849529, 2858216, 2866917, 2875632, 2884361, 2893104, 2901861, 2910632, 2919417, 2928216, 2937029, 2945856, 2954697, 2963552, 2972421, 2981304, 2990201, 2999112, 3008037, 3016976, 3025929, 3034896, 3043877, 3052872, 3061881, 3070904, 3079941, 3088992, 3098057, 3107136, 3116229, 3125336, 3134457, 3143592, 3152741, 3161904, 3171081, 3180272, 3189477, 3198696, 3207929, 3217176, 3226437, 3235712, 3245001, 3254304, 3263621, 3272952, 3282297, 3291656, 3301029, 3310416, 3319817, 3329232, 3338661, 3348104, 3357561, 3367032, 3376517, 3386016, 3395529, 3405056, 3414597, 3424152, 3433721, 3443304, 3452901, 3462512, 3472137, 3481776, 3491429, 3501096, 3510777, 3520472, 3530181, 3539904, 3549641, 3559392, 3569157, 3578936, 3588729, 3598536, 3608357, 3618192, 3628041, 3637904, 3647781, 3657672, 3667577, 3677496, 3687429, 3697376, 3707337, 3717312, 3727301, 3737304, 3747321, 3757352, 3767397, 3777456, 3787529, 3797616, 3807717, 3817832, 3827961, 3838104, 3848261, 3858432, 3868617, 3878816, 3889029, 3899256, 3909497, 3919752, 3930021, 3940304, 3950601, 3960912, 3971237, 3981576, 3991929, 4002296, 4012677, 4023072, 4033481, 4043904, 4054341, 4064792, 4075257, 4085736, 4096229, 4106736, 4117257, 4127792, 4138341, 4148904, 4159481, 4170072, 4180677, 4191296, 4201929, 4212576, 4223237, 4233912, 4244601, 4255304, 4266021, 4276752, 4287497, 4298256, 4309029, 4319816, 4330617, 4341432, 4352261, 4363104, 4373961, 4384832, 4395717, 4406616, 4417529, 4428456, 4439397, 4450352, 4461321, 4472304, 4483301, 4494312, 4505337, 4516376, 4527429, 4538496, 4549577, 4560672, 4571781, 4582904, 4594041, 4605192, 4616357, 4627536, 4638729, 4649936, 4661157, 4672392, 4683641, 4694904, 4706181, 4717472, 4728777, 4740096, 4751429, 4762776, 4774137, 4785512, 4796901, 4808304, 4819721, 4831152, 4842597, 4854056, 4865529, 4877016, 4888517, 4900032, 4911561, 4923104, 4934661, 4946232, 4957817, 4969416, 4981029, 4992656, 5004297, 5015952, 5027621, 5039304, 5051001, 5062712, 5074437, 5086176, 5097929, 5109696, 5121477, 5133272, 5145081, 5156904, 5168741, 5180592, 5192457, 5204336, 5216229, 5228136, 5240057, 5251992, 5263941, 5275904, 5287881, 5299872, 5311877, 5323896, 5335929, 5347976, 5360037, 5372112, 5384201, 5396304, 5408421, 5420552, 5432697, 5444856, 5457029, 5469216, 5481417, 5493632, 5505861, 5518104, 5530361, 5542632, 5554917, 5567216, 5579529, 5591856, 5604197, 5616552, 5628921, 5641304, 5653701, 5666112, 5678537, 5690976, 5703429, 5715896, 5728377, 5740872, 5753381, 5765904, 5778441, 5790992, 5803557, 5816136, 5828729, 5841336, 5853957, 5866592, 5879241, 5891904, 5904581, 5917272, 5929977, 5942696, 5955429, 5968176, 5980937, 5993712, 6006501, 6019304, 6032121, 6044952, 6057797, 6070656, 6083529, 6096416, 6109317, 6122232, 6135161, 6148104, 6161061, 6174032, 6187017, 6200016, 6213029, 6226056, 6239097, 6252152, 6265221, 6278304, 6291401, 6304512, 6317637, 6330776, 6343929, 6357096, 6370277, 6383472, 6396681, 6409904, 6423141, 6436392, 6449657, 6462936, 6476229, 6489536, 6502857, 6516192, 6529541, 6542904, 6556281, 6569672, 6583077, 6596496, 6609929, 6623376, 6636837, 6650312, 6663801, 6677304, 6690821, 6704352, 6717897, 6731456, 6745029, 6758616, 6772217, 6785832, 6799461, 6813104, 6826761, 6840432, 6854117, 6867816, 6881529, 6895256, 6908997, 6922752, 6936521, 6950304, 6964101, 6977912, 6991737, 7005576, 7019429, 7033296, 7047177, 7061072, 7074981, 7088904, 7102841, 7116792, 7130757, 7144736, 7158729, 7172736, 7186757, 7200792, 7214841, 7228904, 7242981, 7257072, 7271177, 7285296, 7299429, 7313576, 7327737, 7341912, 7356101, 7370304, 7384521, 7398752, 7412997, 7427256, 7441529, 7455816, 7470117, 7484432, 7498761, 7513104, 7527461, 7541832, 7556217, 7570616, 7585029, 7599456, 7613897, 7628352, 7642821, 7657304, 7671801, 7686312, 7700837, 7715376, 7729929, 7744496, 7759077, 7773672, 7788281, 7802904, 7817541, 7832192, 7846857, 7861536, 7876229, 7890936, 7905657, 7920392, 7935141, 7949904, 7964681, 7979472, 7994277, 8009096, 8023929, 8038776, 8053637, 8068512, 8083401, 8098304, 8113221, 8128152, 8143097, 8158056, 8173029, 8188016, 8203017, 8218032, 8233061, 8248104, 8263161, 8278232, 8293317, 8308416, 8323529, 8338656, 8353797, 8368952, 8384121, 8399304, 8414501, 8429712, 8444937, 8460176, 8475429, 8490696, 8505977, 8521272, 8536581, 8551904, 8567241, 8582592, 8597957, 8613336, 8628729, 8644136, 8659557, 8674992, 8690441, 8705904, 8721381, 8736872, 8752377, 8767896, 8783429, 8798976, 8814537, 8830112, 8845701, 8861304, 8876921, 8892552, 8908197, 8923856, 8939529, 8955216, 8970917, 8986632, 9002361, 9018104, 9033861, 9049632, 9065417, 9081216, 9097029, 9112856, 9128697, 9144552, 9160421, 9176304, 9192201, 9208112, 9224037, 9239976, 9255929, 9271896, 9287877, 9303872, 9319881, 9335904, 9351941, 9367992, 9384057, 9400136, 9416229, 9432336, 9448457, 9464592, 9480741, 9496904, 9513081, 9529272, 9545477, 9561696, 9577929, 9594176, 9610437, 9626712, 9643001, 9659304, 9675621, 9691952, 9708297, 9724656, 9741029, 9757416, 9773817, 9790232, 9806661, 9823104, 9839561, 9856032, 9872517, 9889016, 9905529, 9922056, 9938597, 9955152, 9971721, 9988304, ...

Jean-Marc a calculé aussi une deuxième suite ; celle composée des côtés des carrés « remplis exactement ».

« Le premier nombre est le terme de la suite [ci-dessus], le second est la taille du côté du carré au coin duquel il se trouve » :

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...

La suite Z_côté des côtés des carrés « pleins » est donc (c’est la succession des deuxièmes nombres dans les accolades ci-dessus) :

Z_côté = 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295, 300, 305, 310, 315, 320, 325, 330, 335, 340, 345, 350, 355, 360, 365, 370, 375, 380, 385, 390, 395, 400, 405, 410, 415, 420, 425, 430, 435, 440, 445, 450, 455, 460, 465, 470, 475, 480, 485, 490, 495, 500, 505, 510, 515, 520, 525, 530, 535, 540, 545, 550, 555, 560, 565, 570, 575, 580, 585, 590, 595, 600, 605, 610, 615, 620, 625, 630, 635, 640, 645, 650, 655, 660, 665, 670, 675, 680, 685, 690, 695, 700, 704, 706, 710, 712, 716, 718, 722, 724, 728, 730, 734, 736, 740, 742, 746, 748, 752, 754, 758, 760, 764, 766, 770, 772, 776, 778, 782, 784, 788, 790, 794, 796, 800, 802, 806, 808, 812, 814, 818, 820, 824, 826, 830, 832, 836, 838, 842, 844, 848, 850, 854, 856, 860, 862, 866, 868, 872, 874, 878, 880, 884, 886, 890, 892, 896, 898, 902, 904, 908, 910, 914, 916, 920, 922, 926, 928, 932, 934, 938, 940, 944, 946, 950, 952, 956, 958, 962, 964, 968, 970, 974, 976, 980, 982, 986, 988, 992, 994, 998, 1000, 1004, 1006, 1010, 1012, 1016, 1018, 1022, 1024, 1028, 1030, 1034, 1036, 1040, 1042, 1046, 1048, 1052, 1054, 1058, 1060, 1064, 1066, 1070, 1072, 1076, 1078, 1082, 1084, 1088, 1090, 1094, 1096, 1100, 1102, 1106, 1108, 1112, 1114, 1118, 1120, 1124, 1126, 1130, 1132, 1136, 1138, 1142, 1144, 1148, 1150, 1154, 1156, 1160, 1162, 1166, 1168, 1172, 1174, 1178, 1180, 1184, 1186, 1190, 1192, 1196, 1198, 1202, 1204, 1208, 1210, 1214, 1216, 1220, 1222, 1226, 1228, 1232, 1234, 1238, 1240, 1244, 1246, 1250, 1252, 1256, 1258, 1262, 1264, 1268, 1270, 1274, 1276, 1280, 1282, 1286, 1288, 1292, 1294, 1298, 1300, 1304, 1306, 1310, 1312, 1316, 1318, 1322, 1324, 1328, 1330, 1334, 1336, 1340, 1342, 1346, 1348, 1352, 1354, 1358, 1360, 1364, 1366, 1370, 1372, 1376, 1378, 1382, 1384, 1388, 1390, 1394, 1396, 1400, 1402, 1406, 1408, 1412, 1414, 1418, 1420, 1424, 1426, 1430, 1432, 1436, 1438, 1442, 1444, 1448, 1450, 1454, 1456, 1460, 1462, 1466, 1468, 1472, 1474, 1478, 1480, 1484, 1486, 1490, 1492, 1496, 1498, 1502, 1504, 1508, 1510, 1514, 1516, 1520, 1522, 1526, 1528, 1532, 1534, 1538, 1540, 1544, 1546, 1550, 1552, 1556, 1558, 1562, 1564, 1568, 1570, 1574, 1576, 1580, 1582, 1586, 1588, 1592, 1594, 1598, 1600, 1604, 1606, 1610, 1612, 1616, 1618, 1622, 1624, 1628, 1630, 1634, 1636, 1640, 1642, 1646, 1648, 1652, 1654, 1658, 1660, 1664, 1666, 1670, 1672, 1676, 1678, 1682, 1684, 1688, 1690, 1694, 1696, 1700, 1702, 1706, 1708, 1712, 1714, 1718, 1720, 1724, 1726, 1730, 1732, 1736, 1738, 1742, 1744, 1748, 1750, 1754, 1756, 1760, 1762, 1766, 1768, 1772, 1774, 1778, 1780, 1784, 1786, 1790, 1792, 1796, 1798, 1802, 1804, 1808, 1810, 1814, 1816, 1820, 1822, 1826, 1828, 1832, 1834, 1838, 1840, 1844, 1846, 1850, 1852, 1856, 1858, 1862, 1864, 1868, 1870, 1874, 1876, 1880, 1882, 1886, 1888, 1892, 1894, 1898, 1900, 1904, 1906, 1910, 1912, 1916, 1918, 1922, 1924, 1928, 1930, 1934, 1936, 1940, 1942, 1946, 1948, 1952, 1954, 1958, 1960, 1964, 1966, 1970, 1972, 1976, 1978, 1982, 1984, 1988, 1990, 1994, 1996, 2000, 2002, 2006, 2008, 2012, 2014, 2018, 2020, 2024, 2026, 2030, 2032, 2036, 2038, 2042, 2044, 2048, 2050, 2054, 2056, 2060, 2062, 2066, 2068, 2072, 2074, 2078, 2080, 2084, 2086, 2090, 2092, 2096, 2098, 2102, 2104, 2108, 2110, 2114, 2116, 2120, 2122, 2126, 2128, 2132, 2134, 2138, 2140, 2144, 2146, 2150, 2152, 2156, 2158, 2162, 2164, 2168, 2170, 2174, 2176, 2180, 2182, 2186, 2188, 2192, 2194, 2198, 2200, 2204, 2206, 2210, 2212, 2216, 2218, 2222, 2224, 2228, 2230, 2234, 2236, 2240, 2242, 2246, 2248, 2252, 2254, 2258, 2260, 2264, 2266, 2270, 2272, 2276, 2278, 2282, 2284, 2288, 2290, 2294, 2296, 2300, 2302, 2306, 2308, 2312, 2314, 2318, 2320, 2324, 2326, 2330, 2332, 2336, 2338, 2342, 2344, 2348, 2350, 2354, 2356, 2360, 2362, 2366, 2368, 2372, 2374, 2378, 2380, 2384, 2386, 2390, 2392, 2396, 2398, 2402, 2404, 2408, 2410, 2414, 2416, 2420, 2422, 2426, 2429, 2436, 2443, 2450, 2457, 2464, 2471, 2478, 2485, 2492, 2499, 2506, 2513, 2520, 2527, 2534, 2541, 2548, 2555, 2562, 2569, 2576, 2583, 2590, 2597, 2604, 2611, 2618, 2625, 2632, 2639, 2646, 2653, 2660, 2667, 2674, 2681, 2688, 2695, 2702, 2709, 2716, 2723, 2730, 2737, 2744, 2751, 2758, 2765, 2772, 2779, 2786, 2793, 2800, 2807, 2814, 2821, 2828, 2835, 2842, 2849, 2856, 2863, 2870, 2877, 2884, 2891, 2898, 2905, 2912, 2919, 2926, 2933, 2940, 2947, 2954, 2961, 2968, 2975, 2982, 2989, 2996, 3003, 3010, 3017, 3024, 3031, 3038, 3045, 3052, 3059, 3066, 3073, 3080, 3087, 3094, 3101, 3108, 3115, 3122, 3129, 3136, 3143, 3150, 3157, 3164, 3171, 3178, 3185, 3192, 3199, 3206, 3213, 3220, 3227, 3234, 3241, 3248, 3255, 3262, 3269, 3276, 3283, 3290, 3297, 3304, 3311, 3318, 3325, 3332, 3339, 3346, 3353, 3360, 3367, 3374, 3381, 3388, 3395, 3402, 3409, 3416, 3423, 3430, 3437, 3444, 3451, 3458, 3465, 3472, 3479, 3486, 3493, 3500, 3507, 3514, 3521, 3528, 3535, 3542, 3549, 3556, 3563, 3570, 3577, 3584, 3591, 3598, 3605, 3612, 3619, 3626, 3633, 3640, 3647, 3654, 3661, 3668, 3675, 3682, 3689, 3696, 3703, 3710, 3717, 3724, 3731, 3738, 3745, 3752, 3759, 3766, 3773, 3780, 3787, 3794, 3801, 3808, 3815, 3822, 3829, 3836, 3843, 3850, 3857, 3864, 3871, 3878, 3885, 3892, 3899, 3906, 3913, 3920, 3927, 3934, 3941, 3948, 3955, 3962, 3969, 3976, 3983, 3990, 3997, 4004, 4011, 4018, 4025, 4032, 4039, 4046, 4053, 4060, 4067, 4074, 4081, 4088, 4095, 4102, 4109, 4116, 4123, 4130, 4137, 4144, 4151, 4158, 4165, 4172, 4179, 4186, 4193, 4200, 4207, 4214, 4221, 4228, 4235, 4242, 4249, 4256, 4263, 4270, 4277, 4284, 4291, 4298, 4305, 4312, 4319, 4326, 4333, 4340, 4347, 4354, 4361, 4368, 4375, 4382, 4389, 4396, 4403, 4410, 4417, 4424, 4431, 4438, 4445, 4452, 4459, 4466, 4473, 4480, 4487, 4494, 4501, 4508, 4515, 4522, 4529, 4536, 4543, 4550, 4557, 4564, 4571, 4578, 4585, 4592, 4599, 4606, 4613, 4620, 4627, 4634, 4641, 4648, 4655, 4662, 4669, 4676, 4683, 4690, 4697, 4704, 4711, 4718, 4725, 4732, 4739, 4746, 4753, 4760, 4767, 4774, 4781, 4788, 4795, 4802, 4809, 4816, 4823, 4830, 4837, 4844, 4851, 4858, 4865, 4872, 4879, 4886, 4893, 4900, 4907, 4914, 4921, 4928, 4935, 4942, 4949, 4956, 4963, 4970, 4977, 4984, 4991, 4998, 5005, 5012, 5019, 5026, 5033, 5040, 5047, 5054, 5061, 5068, 5075, 5082, 5089, 5096, 5103, 5110, 5117, 5124, 5131, 5138, 5145, 5152, 5159, 5166, 5173, 5180, 5187, 5194, 5201, 5208, 5215, 5222, 5229, 5236, 5243, 5250, 5257, 5264, 5271, 5278, 5285, 5292, 5299, 5306, 5313, 5320, 5327, 5334, 5341, 5348, 5355, 5362, 5369, 5376, 5383, 5390, 5397, 5404, 5411, 5418, 5425, 5432, 5439, 5446, 5453, 5460, 5467, 5474, 5481, 5488, 5495, 5502, 5509, 5516, 5523, 5530, 5537, 5544, 5551, 5558, 5565, 5572, 5579, 5586, 5593, 5600, 5607, 5614, 5621, 5628, 5635, 5642, 5649, 5656, 5663, 5670, 5677, 5684, 5691, 5698, 5705, 5712, 5719, 5726, 5733, 5740, 5747, 5754, 5761, 5768, 5775, 5782, 5789, 5796, 5803, 5810, 5817, 5824, 5831, 5838, 5845, 5852, 5859, 5866, 5873, 5880, 5887, 5894, 5901, 5908, 5915, 5922, 5929, 5936, 5943, 5950, 5957, 5964, 5971, 5978, 5985, 5992, 5999, 6006, 6013, 6020, 6027, 6034, 6041, 6048, 6055, 6062, 6069, 6076, 6083, 6090, 6097, 6104, 6111, 6118, 6125, 6132, 6139, 6146, 6153, 6160, 6167, 6174, 6181, 6188, 6195, 6202, 6209, 6216, 6223, 6230, 6237, 6244, 6251, 6258, 6265, 6272, 6279, 6286, 6293, 6300, 6307, 6314, 6321, 6328, 6335, 6342, 6349, 6356, 6363, 6370, 6377, 6384, 6391, 6398, 6405, 6412, 6419, 6426, 6433, 6440, 6447, 6454, 6461, 6468, 6475, 6482, 6489, 6496, 6503, 6510, 6517, 6524, 6531, 6538, 6545, 6552, 6559, 6566, 6573, 6580, 6587, 6594, 6601, 6608, 6615, 6622, 6629, 6636, 6643, 6650, 6657, 6664, 6671, 6678, 6685, 6692, 6699, 6706, 6713, 6720, 6727, 6734, 6741, 6748, 6755, 6762, 6769, 6776, 6783, 6790, 6797, 6804, 6811, 6818, 6825, 6832, 6839, 6846, 6853, 6860, 6867, 6874, 6881, 6888, 6895, 6902, 6909, 6916, 6923, 6930, 6937, 6944, 6951, 6958, 6965, 6972, 6979, 6986, 6993, 7000, 7007, 7014, 7021, 7028, 7035, 7042, 7049, 7056, 7063, 7070, 7077, 7084, 7091, 7098, 7105, 7112, 7119, 7126, 7133, 7140, 7147, 7154, 7161, 7168, 7175, 7182, 7189, 7196, 7203, 7210, 7217, 7224, 7231, 7238, 7245, 7252, 7259, 7266, 7273, 7280, 7287, 7294, 7301, 7308, 7315, 7322, 7329, 7336, 7343, 7350, 7357, 7364, 7371, 7378, 7385, 7392, 7399, 7406, 7413, 7420, 7427, 7434, 7441, 7448, 7455, 7462, 7469, 7476, 7483, 7490, 7497, 7504, 7511, 7518, 7525, 7532, 7539, 7546, 7553, 7560, 7567, 7574, 7581, 7588, 7595, 7602, 7609, 7616, 7623, 7630, 7637, 7644, 7651, 7658, 7665, 7672, 7679, 7686, 7693, 7700, 7707, 7714, 7721, 7728, 7735, 7742, 7749, 7756, 7763, 7770, 7777, 7784, 7791, 7798, 7805, 7812, 7819, 7826, 7833, 7840, 7847, 7854, 7861, 7868, 7875, 7882, 7889, 7896, 7903, 7910, 7917, 7924, 7931, 7938, 7945, 7952, 7959, 7966, 7973, 7980, 7987, 7994, 8001, 8008, 8015, 8022, 8029, 8036, 8043, 8050, 8057, 8064, 8071, 8078, 8085, 8092, 8099, 8106, 8113, 8120, 8127, 8134, 8141, 8148, 8155, 8162, 8169, 8176, 8183, 8190, 8197, 8204, 8211, 8218, 8225, 8232, 8239, 8246, 8253, 8260, 8267, 8274, 8281, 8288, 8295, ...

Merci et bravo à Jean-Marc pour ces calculs (et à Paolo Lava pour ses remarques – je lui dois « ... and leaving no “hole” », dans l’explication tout en haut de la page ; merci aussi à Joshua Zucker pour avoir attiré mon attention sur les carrés qui se remplissent aussi par le coin N.-E., ce que j’avais négligé).

Au lieu de disposer ainsi les chiffres des entiers, on aurait pu « spiraler » les chiffres des nombres premiers, des nombres de Fibonacci, etc. – et ne retenir que les nombres dont les chiffres, disposés un à un en « spirale carrée », remplissent exactement un carré. Les listes de Jean-Marc figurent ci-dessous.

Suite Pr des nombres premiers dont les chiffres, disposés en spirale carrée (dans le sens des aiguilles d’une montre à partir de « 2 »), remplissent exactement un carré :

Spirale Suite Pr Côté du carré « plein »

= suite Pr_côté

2 -> 2 1

2 3

7 5 -> 7 2

1 3 1 7

1 2 3 1

1 7 5 9

9 2 3 2 -> 29 4

... ... ...

Pr = 2, 7, 29, 71, 101, 127, 191, 229, 317, 379, 499, 577, 733, 823, 10867, 11159, 12301, 12577, 13781, 14107, 15391, 15733, 17183, 17509, 19079, 19457, 21023, 21467, 23059, 23549, 25339, 25793, 27733, 28151, 30161, 30697, 32719, 33247, 35401, 35993, 38273, 38861, 41233, 41851, 44267, 44909, 47591, 48259, 50923, 51577, 54403, 55079, 57859, 58601, 61643, 62467, 65587, 66403, 69691, 70481, 73709, 74561, 77951, 78857, 82279, 83231, 86951, 87803, 91583, 92551, 96269, 97301, 101891, 103643, 107021, 108877, 112337, 114199, 117839, 119747, 123419, 125287, 129119, 131111, 135151, 136991, 141079, 143197, 147419, 149503, 153719, 155833, 160403, 162683, 167221, 169681, 174221, 176549, 181301, 183637, 188459, 190891, 195893, 198391, 203563, 206179, 211151, 213887, 219281, 221873, 227363, 229897, 235591, 238499, 244159, 246919, 252641, 255551, 261593, 264619, 270407, 273433, 279637, 282671, 289099, 292181, 298733, 301927, 308317, 311681, 318107, 321289, 327839, 331241, 338027, 341569, 348437, 351919, 358909, 362693, 369959, 373657, 380929, 384599, 392033, 395537, 403327, 407317, 414857, 418739, 426551, 430433, 438091, 442109, 450137, 454247, 462653, 466729, 474757, 478823, 487397, 491483, 499819, 504247, 512921, 517189, 525769, 530177, 539107, 543607, 552731, 557269, 566453, 570967, 580187, 584849, 594119, 598783, 608213, 612929, 622493, 627479, 637283, 642457, 652321, 657233, 667127, 672169, 682207, 687311, 697447, 702707, 712837, 718259, 728647, 733871, 744659, 750131, 760807, 766021, 776969, 782191, 793187, 798533, 809803, 815413, 826351, 832003, 843503, 849223, 860861, 866471, 878083, 883763, 895507, 901501, 913247, 919313, 931237, 937337, 949633, 955709, 968041, 974147, 986729, 993103, 1002017, 1020599, 1039891, 1059299, 1079059, 1098593, 1118149, 1138591, 1159001, 1179323, 1199899, 1221239, 1243013, 1264331, 1285981, 1308157, 1330393, 1352963, 1375681, 1398151, 1421737, 1444789, 1468109, 1491821, 1515571, 1540151, 1564721, 1589591, 1614631, 1639723, 1665233, 1690669, 1716389, 1742731, 1769129, 1795537, 1822321, 1849103, 1875983, 1903463, 1930633, 1958699, 1987217, 2015021, 2043491, 2072431, 2101721, 2130917, 2160001, 2190031, 2220403, 2250649, 2280451, 2311237, 2342671, 2373929, 2405239, 2437037, 2468777, 2501099, 2533007, 2566019, 2598157, 2631427, 2664307, 2697823, 2731621, 2765767, 2800001, 2833723, 2868181, 2903671, 2938697, 2974513, 3009859, 3045989, 3081919, 3118211, 3154597, 3191351, 3227821, 3265021, 3302749, 3340453, 3378499, 3416509, 3454687, 3494363, 3533897, 3573337, 3612283, 3652223, 3691657, 3731107, 3772259, 3812773, 3853321, 3894227, 3935761, 3977387, 4019731, 4061227, 4103543, 4146811, 4188973, 4231943, 4275329, 4319299, 4363487, 4407827, 4450967, 4495061, 4540541, 4585297, 4630447, 4676101, 4721531, 4767533, 4814123, 4860463, 4906483, 4953391, 5000077, 5047843, 5095361, 5143349, 5192023, 5240107, 5288687, 5337463, 5386351, 5436163, 5485633, 5535773, 5586227, 5636783, 5688077, 5738633, 5790157, 5841707, 5892949, 5943919, 5996633, 6049171, 6101621, 6154339, 6207569, 6261121, 6313961, 6367853, 6422677, 6477277, 6531619, 6586201, 6641381, 6697241, 6752177, 6808097, 6863629, 6919699, 6975883, 7032787, 7089553, 7147223, 7204849, 7262819, 7320697, 7378993, 7438037, 7497143, 7556239, 7615477, 7674829, 7734827, 7795687, 7855273, 7915643, 7976651, 8038483, 8099963, 8161511, 8222969, 8284937, 8347253, 8410301, 8473793, 8535991, 8599037, 8662127, 8725961, 8789579, 8854487, 8919403, 8984351, 9049973, 9115663, 9181141, 9246701, 9312593, 9379049, 9446321, 9513113, 9580387, 9647317, 9715241, 9783373, 9851563, 9920467, 9989923, 10008571, 10025497, 10042757, 10059703, 10077071, 10094281, 10111643, 10129003, 10146121, 10163561, 10181251, 10199131, 10216849, 10233911, 10251671, 10268617, 10286257, 10304033, 10321691, 10338787, 10356721, 10374037, 10391873, 10409249, 10426733, 10444331, 10461887, 10479341, 10496923, 10514437, 10532647, 10550191, 10567717, 10585741, 10603841, 10621777, 10639571, 10657321, 10674689, 10692323, 10710103, 10727947, 10746277, 10764311, 10782517, 10800767, 10818683, 10836913, 10854821, 10872703, 10891253, 10909081, 10926989, 10945057, 10963097, 10981591, 10999871, 11018549, 11036903, 11054783, 11073047, 11091127, 11109151, 11127997, 11146637, 11164553, 11182679, 11201093, 11219717, 11238401, 11256467, 11275399, 11293669, 11312557, 11331121, 11349803, 11368531, 11387237, 11405969, 11424473, 11443379, 11461609, 11480363, 11499209, 11518271, 11536673, 11554819, 11573803, 11592563, 11610901, 11629337, 11648873, 11667301, 11686523, 11705269, 11724473, 11743477, 11762477, 11781619, 11800253, 11819179, 11838487, 11857877, 11876701, 11895799, 11914883, 11933771, 11952569, 11971819, 11990827, 12010511, 12029737, 12049109, 12067933, 12087617, 12107129, 12126349, 12145379, 12165011, 12184493, 12204169, 12223199, 12242501, 12262121, 12281483, 12300713, 12319799, 12339823, 12359317, 12378199, 12397787, 12417007, 12436651, 12456371, 12476221, 12495971, 12515903, 12535559, 12555293, 12575417, 12595103, 12614653, 12634067, 12654221, 12674297, 12693893, 12713777, 12733129, 12752911, 12772481, 12792547, 12812413, 12832241, 12852139, 12871763, 12892459, 12912863, 12933127, 12953323, 12973529, 12994027, 13013743, 13034267, 13054267, 13075043, 13095703, 13115929, 13136117, 13156471, 13176841, 13196873, 13216991, 13237267, 13258103, 13278329, 13298867, 13318757, 13339357, 13360007, 13380137, 13400479, 13421717, 13441943, 13462919, 13483633, 13504033, 13525133, 13545907, 13566571, 13586999, 13607411, 13628177, 13648823, 13669013, 13689551, 13710413, 13731173, 13752017, 13773299, 13793347, 13814419, 13835389, 13855819, 13876763, 13897337, 13918049, 13939249, 13960279, 13981921, 14003153, 14024207, 14045231, 14066581, 14087537, 14108467, 14129299, 14150509, 14171753, 14193601, 14214791, 14235779, 14257073, 14278091, 14299163, 14320351, 14341837, 14362351, 14383693, 14404879, 14426231, 14447737, 14469487, 14490659, 14512403, 14534041, 14555491, 14576761, 14597819, 14619893, 14641831, 14663659, 14684743, 14706709, 14728291, 14749841, 14771201, 14792543, 14814409, 14836273, 14857819, 14879737, 14901823, 14923921, 14945573, 14967791, 14989967, 15011693, 15033521, 15055343, 15077269, 15099341, 15121537, 15143657, 15165299, 15187693, 15209963, 15231913, 15253817, 15275581, 15297649, 15319651, 15342167, 15363923, 15386981, 15409663, 15431959, 15454427, 15476569, ...

Suite Pr_côté des côtés des carrés remplis exactement par les chiffres des nombres premiers ci-dessus :

Pr_côté = 1, 2, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 72, 73, 77, 78, 82, 83, 87, 88, 92, 93, 97, 98, 102, 103, 107, 108, 112, 113, 117, 118, 122, 123, 127, 128, 132, 133, 137, 138, 142, 143, 147, 148, 152, 153, 157, 158, 162, 163, 167, 168, 172, 173, 177, 178, 182, 183, 187, 188, 192, 193, 197, 198, 202, 203, 207, 208, 212, 213, 218, 220, 224, 226, 230, 232, 236, 238, 242, 244, 248, 250, 254, 256, 260, 262, 266, 268, 272, 274, 278, 280, 284, 286, 290, 292, 296, 298, 302, 304, 308, 310, 314, 316, 320, 322, 326, 328, 332, 334, 338, 340, 344, 346, 350, 352, 356, 358, 362, 364, 368, 370, 374, 376, 380, 382, 386, 388, 392, 394, 398, 400, 404, 406, 410, 412, 416, 418, 422, 424, 428, 430, 434, 436, 440, 442, 446, 448, 452, 454, 458, 460, 464, 466, 470, 472, 476, 478, 482, 484, 488, 490, 494, 496, 500, 502, 506, 508, 512, 514, 518, 520, 524, 526, 530, 532, 536, 538, 542, 544, 548, 550, 554, 556, 560, 562, 566, 568, 572, 574, 578, 580, 584, 586, 590, 592, 596, 598, 602, 604, 608, 610, 614, 616, 620, 622, 626, 628, 632, 634, 638, 640, 644, 646, 650, 652, 656, 658, 662, 664, 668, 670, 674, 676, 679, 686, 693, 700, 707, 714, 721, 728, 735, 742, 749, 756, 763, 770, 777, 784, 791, 798, 805, 812, 819, 826, 833, 840, 847, 854, 861, 868, 875, 882, 889, 896, 903, 910, 917, 924, 931, 938, 945, 952, 959, 966, 973, 980, 987, 994, 1001, 1008, 1015, 1022, 1029, 1036, 1043, 1050, 1057, 1064, 1071, 1078, 1085, 1092, 1099, 1106, 1113, 1120, 1127, 1134, 1141, 1148, 1155, 1162, 1169, 1176, 1183, 1190, 1197, 1204, 1211, 1218, 1225, 1232, 1239, 1246, 1253, 1260, 1267, 1274, 1281, 1288, 1295, 1302, 1309, 1316, 1323, 1330, 1337, 1344, 1351, 1358, 1365, 1372, 1379, 1386, 1393, 1400, 1407, 1414, 1421, 1428, 1435, 1442, 1449, 1456, 1463, 1470, 1477, 1484, 1491, 1498, 1505, 1512, 1519, 1526, 1533, 1540, 1547, 1554, 1561, 1568, 1575, 1582, 1589, 1596, 1603, 1610, 1617, 1624, 1631, 1638, 1645, 1652, 1659, 1666, 1673, 1680, 1687, 1694, 1701, 1708, 1715, 1722, 1729, 1736, 1743, 1750, 1757, 1764, 1771, 1778, 1785, 1792, 1799, 1806, 1813, 1820, 1827, 1834, 1841, 1848, 1855, 1862, 1869, 1876, 1883, 1890, 1897, 1904, 1911, 1918, 1925, 1932, 1939, 1946, 1953, 1960, 1967, 1974, 1981, 1988, 1995, 2002, 2009, 2016, 2023, 2030, 2037, 2044, 2051, 2058, 2065, 2072, 2079, 2086, 2093, 2100, 2107, 2114, 2121, 2128, 2135, 2137, 2139, 2141, 2143, 2145, 2147, 2149, 2151, 2153, 2155, 2157, 2159, 2161, 2163, 2165, 2167, 2169, 2171, 2173, 2175, 2177, 2179, 2181, 2183, 2185, 2187, 2189, 2191, 2193, 2195, 2197, 2199, 2201, 2203, 2205, 2207, 2209, 2211, 2213, 2215, 2217, 2219, 2221, 2223, 2225, 2227, 2229, 2231, 2233, 2235, 2237, 2239, 2241, 2243, 2245, 2247, 2249, 2251, 2253, 2255, 2257, 2259, 2261, 2263, 2265, 2267, 2269, 2271, 2273, 2275, 2277, 2279, 2281, 2283, 2285, 2287, 2289, 2291, 2293, 2295, 2297, 2299, 2301, 2303, 2305, 2307, 2309, 2311, 2313, 2315, 2317, 2319, 2321, 2323, 2325, 2327, 2329, 2331, 2333, 2335, 2337, 2339, 2341, 2343, 2345, 2347, 2349, 2351, 2353, 2355, 2357, 2359, 2361, 2363, 2365, 2367, 2369, 2371, 2373, 2375, 2377, 2379, 2381, 2383, 2385, 2387, 2389, 2391, 2393, 2395, 2397, 2399, 2401, 2403, 2405, 2407, 2409, 2411, 2413, 2415, 2417, 2419, 2421, 2423, 2425, 2427, 2429, 2431, 2433, 2435, 2437, 2439, 2441, 2443, 2445, 2447, 2449, 2451, 2453, 2455, 2457, 2459, 2461, 2463, 2465, 2467, 2469, 2471, 2473, 2475, 2477, 2479, 2481, 2483, 2485, 2487, 2489, 2491, 2493, 2495, 2497, 2499, 2501, 2503, 2505, 2507, 2509, 2511, 2513, 2515, 2517, 2519, 2521, 2523, 2525, 2527, 2529, 2531, 2533, 2535, 2537, 2539, 2541, 2543, 2545, 2547, 2549, 2551, 2553, 2555, 2557, 2559, 2561, 2563, 2565, 2567, 2569, 2571, 2573, 2575, 2577, 2579, 2581, 2583, 2585, 2587, 2589, 2591, 2593, 2595, 2597, 2599, 2601, 2603, 2605, 2607, 2609, 2611, 2613, 2615, 2617, 2619, 2621, 2623, 2625, 2627, 2629, 2631, 2633, 2635, 2637, 2639, 2641, 2643, 2645, 2647, 2649, 2651, 2653, 2655, 2657, 2659, 2661, 2663, 2665, 2667, 2669, 2671, 2673, 2675, 2677, 2679, 2681, 2683, 2685, 2687, 2689, 2691, ...

Faisons de même avec les nombres de Fibonacci (attention, c’est du brutal !) :

Spirale Suite Fi Côté du carré « plein »

= suite Fi_côté

0 -> 0 1

0 1

2 1 -> 2 2

8 1 3 -> 13 3

5 0 1

3 2 1

2 3 3 3 7 7

4 8 1 3 2 6

4 5 0 1 1 1

1 3 2 1 3 0

9 8 5 5 4 9

7 9 5 1 7 8 -> 1597 6

... ... ...

Fi = 0, 2, 13, 1597, 1134903170, 3416454622906707, 10284720757613717413913, 394810887814999156320699623170776339, 141530751622060734789349637541611806906560581814825656065057782655897254318057662142341135314844769422903905867863877139246681886097354486547763701, 50735565489499797512576144344242780042620227238147640546853218443231457841287797456600083212822748558767337492604319704628923840439204144749031679717828456000580783593116489752032700480117938528975177935704132468083995866400821866352373075439900593592513640, 9753147955551871675612234711772179164748733799966446876860847418296892311792942831984522609729861198863720764069198578765852965083707518649064787886536243319977702982625877912277912899805792399732018143179599897466315018977626304698986477384078235400525697809212706426817955451596994443293581716277589298722686567010849582040001850818556935252598867628950692271862772741780205586539291274432629757684216508159374510572183852279477766187745475449607299590606046916519565258293987965378493692373714081088766075943244117130198115587420734354127280706, 7142168961210661162436558623109220850442785675017083549650127758724432220878152069539868803042442359352940354388753622776519118295089465227362538112055298579826991443190211780676669663173340918355074262469549490432084367034963976566112452361223938895127381379408022634381313107578776652687750153393799605202801615518275190993510883261536258905977845042711255407426523584772660267289719676404173796927714040234838271215556064256382634454198209178994863214248075782281002669755820878887283678132920378289779546022649429356701936664973674007008715102261791730548267903141836829273422554669872023605159159382877436406815850179734993431640049, 736263112118010383289065385806094353294344467866877106901089406197774358987324156238789502513921754158484694668917631545963101448483162226024667201258807806425034328700276221838585720370549013501600503052473680262468168893310096369041846362496170032807499026620913942718153228745888426730253205160351731341102122361363535611863055415043487422893289441191454976415776420349979169590360444689479736969751658546406832991194730075750915141838790907508322195583038686846450820750826769942000026943118132563599353014225874548480381615115729695233950545222353638882633837385647474121733489662933528979144285500646288206977742361679163865435448356305999665996689307896778639031505260463091964705791904448483529933002884327467592198819374858246258954067094708843050225958621194990451075373687882100454800170816017729712171026731705316800312008920720205606683185631684014680888448021601176517398796199130531762353107257031299893562594133447723346317133933533885679488113767613778081201932212411535737571234142005868953024722865032349132793423830139286462006225018333306435820043369938059308041966812121370610950000024, 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...

Suite Fi_côté des côtés des carrés remplis exactement par les chiffres des nombres de Fibonacci ci-dessus :

Fi_côté = 1, 2, 3, 6, 15, 25, 35, 56, 227, 398, 847, 986, 1713, 4589, 6460, 7465, 24860, 28741, ...

__________

Au lieu de semer les chiffres des « nonnegative integers » (soit l’ensemble Z+ des nombres entiers), on aurait pu disposer de même les chiffres des carrés parfaits. Une sorte de mise en abyme : des carrés semés dans des carrés ! Plus besoin de « spiraler », il suffit de remplir successivement les bordures est et sud du carré qui précède, en commençant avec un carré unité. On obtient :

Côté du carré : 1 2 4 5 7 8 ...

0|1|1 6|1|1 6|4|6 1 9

4 9|6 4|0|6 2|4|7 1 6

2 5 3 9|0|9 8|1|6 0 .

6 4 8 1|1|1 9|4|7 2 . E

2 1 1 4 4|9 3|8|2 4 . S

6 2 2 5 2 5 2|4|9 1 T

4 3 6 1 4 0 0|5|7 0

2 9 5 7 6 6 2 5|8 8

4 8 4 1 9 0 0 9 6 9

1 1 5 6 1 2 2 5 1 2

S U D

C_abyme = 0, 9, 81, 144, 400, 625, ... (le dernier chiffre d’un nombre jaune – ici en gras – est sur la diagonale descendante \ du carré).

CoC_abyme = 1, 2, 4, 5, 7, 8, ... (numéro des colonnes comportant un chiffre gras ou côté du carré)

Jean-Marc a calculé ces deux suites (ainsi que toutes celles plus bas) :

C_abyme = 0, 9, 81, 144, 400, 625, 15625, 17956, 30276, 34225, 54289, 60516, 91204, 1113025, 1478656, 1934881, 2496400, 3179089, 4000000, 4977361, 6130576, 7480225, 9048064, 1009205824, 1063281664, 1077152400, 1134072976, 1148667664, 1208535696, 1223880256, 1286800384, 1302921216, 1369000000, 1385923984, 1455269904, 1473024400, 1545747856, 1564360704, 1640574016, 1660073536, 1739890944, 1760305936, 1843843600, 1865203344, 1952579344, 1974913600, 2066247936, 2089586944, 2185001536, 2209376016, 2308994704, 2334435856, 2438384400, 2464923904, ...

CoC_abyme = 1, 2, 4, 5, 7, 8, 22, 23, 27, 28, 32, 33, 37, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 521, 529, 531, 539, 541, 549, 551, 559, 561, 569, 571, 579, 581, 589, 591, 599, 601, 609, 611, 619, 621, 629, 631, 639, 641, 649, 651, 659, 661, 669, 671, ...

Notons que le remplissage par bordures est et sud est équivalent au remplissage d’un escalier où les marches s’allongent de deux unités à chaque fois :

*

* * *

* * * * *

* * * * * * *

* * * * * * * * *

* * * * * * * * * * *

* * * * * * * * * * * * * <-- deux unités (deux cases) supplémentaires pour chaque marche

...

On remplit ainsi (on sème la succession des chiffres des carrés parfaits dans chaque case et on garde les nombres « qui tombent juste » en bout de marche) :

0

1,4,9

1 6,2 5,3

6,4 9,6 4,8 1

1 0 0,1 2 1,1 4 4

1 6 9,1 9 6,2 2 5,2 5

6,2 8 9,3 2 4,3 6 1,4 0 0

4 4 1,4 8 4,5 2 9,5 7 6,6 2 5

...

Tant qu’à jouer la mise en abyme, disposons en triangle les chiffres des nombres triangulaires – et conservons ceux « qui tombent juste » (ils forment la suite Tri_abyme) ; les marches de l’escalier grandissent ici d’une unité à chaque fois (la suite LiTri_abyme reprend les numéros des lignes du triangle comportant un nombre en jaune) :

0

1,3

6,1 0

1 5,2 1

2 8,3 6,4

5,5 5,6 6,7

8,9 1,1 0 5,1

2 0,1 3 6,1 5 3

1 7 1,1 9 0,2 1 0

2 3 1,2 5 3,2 7 6,3

0 0,3 2 5,3 5 1,3 7 8

4 0 6,4 3 5,4 6 5,4 9 6

5 2 8,5 6 1,5 9 5,6 3 0,6

6 6,7 0 3,7 4 1,7 8 0,8 2 0

8 6 1,9 0 3,9 4 6,9 9 0,1 0 3

5,1 0 8 1,1 1 2 8,1 1 7 6,1 2 2

5,1 2 7 5,1 3 2 6,1 3 7 8,1 4 3 1

...

Tri_abyme = 0, 3, 10, 21, 153, 210, 378, 496, 820, 1431, 3081, 4656, 8646, 11628, 15051, 17766, 22578, 26335, 32896, 37950, 46665, 53301, 64620, 73153, 87571, 98346, 108345, 113526, 130305, 162735, 185136, 193131, 218791, 267546, 300700, 312445, 349866, 419986, 467061, 483636, 536130, 633375, 697971, 720600, 791911, 922761, 1007490, 1033203, 1198926, 1228528, 1418770, 1452660, 1669878, 1708476, 1955253, 1999000, 2278045, 2327403, 2641551, 2697003, 3049215, 3111265, 3504628, 3573801, 4011528, 4088370, 4573800, 4658878, 5195476, 5289378, 5880735, 5984070, 6633903, 6747301, 7459453, 7583565, 8362005, 8497503, 9346326, 9493903, 10504236, 11226691, 12956595, 13810140, 15845635, 16846110, 19223100, 20387305, 23143806, 24489501, 27665641, 29211546, 32849565, 34615360, 38759610, 40765935, 45462880, 47731335, ...

LiTri_abyme = 1, 2, 3, 4, 8, 9, 11, 12, 14, 17, 22, 25, 30, 33, 36, 38, 41, 43, 46, 48, 51, 53, 56, 58, 61, 63, 65, 66, 69, 74, 77, 78, 81, 86, 89, 90, 93, 98, 101, 102, 105, 110, 113, 114, 117, 122, 125, 126, 132, 133, 139, 140, 146, 147, 153, 154, 160, 161, 167, 168, 174, 175, 181, 182, 188, 189, 195, 196, 202, 203, 209, 210, 216, 217, 223, 224, 230, 231, 237, 238, 245, 250, 261, 266, 277, 282, 293, 298, 309, 314, 325, 330, 341, 346, 357, 362, 373, 378, ...

Mise en abyme des nombres pentagonaux — on sème les chiffres des pentagonaux sur les marches de l’escalier suivant :

*

* * * *

* * * * * * *

* * * * * * * * * *

* * * * * * * * * * * * * <-- la marche s’allonge de trois cases à chaque fois

P_abyme = 0, 12, 590, 1001, 5551, 11837, 17442, 29751, 41251, 64792, 85562, 109215, 264390, 343922, 699392, 866020, 1114135, 1316485, 1631252, 1904630, 2324415, 2684697, 3231802, 3696565, 4395560, 4984082, 5861805, 6595065, 7680622, 8581300, 9906065, 12622251, 15444917, 19785320, 23782495, 29821792, 35295876, 43446195, 50735876, 61456001, 70936255, 84731626, 96813717, 103804482, 120068740, 138292805, 158635842, 181263577, 206348297, 234068850, 264610645, 298165652, 334932402, 375115987, 418928060, 466586835, 518317087, 574350152, 634923927, 700282870, 770678000, 846366897, 927613702, ...

LiP_abyme = 1, 2, 6, 7, 12, 15, 17, 20, 22, 25, 27, 29, 38, 41, 50, 53, 57, 60, 64, 67, 71, 74, 78, 81, 85, 88, 92, 95, 99, 102, 106, 114, 121, 130, 137, 146, 153, 162, 169, 178, 185, 194, 201, 205, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 304, 313, 322, 331, 340, 349, 358, 367, 376, ...

Mise en abyme des nombres hexagonaux — on sème les chiffres des hexagonaux sur les marches de l’escalier suivant :

*

* * * * *

* * * * * * * * *

* * * * * * * * * * * * *

* * * * * * * * * * * * * * * * * < -- la marche s’allonge de quatre cases à chaque fois

H_abyme = 0, 276, 1128, 5995, 14365, 22791, 42486, 61776, 132870, 271953, 507528, 878475, 1558495, 2418900, 3616705, 5234230, 7363203, 11729746, 15643621, 20528028, 26539255, 33845878, 42628761, 53081056, 65408203, 79827930, 96570253, 1112952610, 1238501565, 1374950580, 1522940455, 1683131190, 1856201985, 2042851240, 2243796555, 2459774730, 2691541765, 2939872860, 3205562415, 3489424030, 3792290505, 4115013840, 4458465235, 4823535090, ...

LiH_abyme = 1, 4, 6, 10, 13, 15, 18, 20, 25, 31, 37, 43, 51, 58, 65, 72, 79, 90, 98, 106, 114, 122, 130, 138, 146, 154, 162, 319, 329, 339, 349, 359, 369, 379, 389, 399, 409, 419, 429, 439, 449, 459, 469, 479, ...

Mise en abyme des nombres heptagonaux (même technique de semis que supra) :

7_abyme = 0, 34, 1782, 7209, 10465, 39501, 109935, 172528, 338008, 485541, 544522, 755425, 11116539, 11617606, 21545836, 22390633, 38468938, 39794265, 64254645, 66221302, 1031108239, 1247990608, 1327760721, 1591344405, 1687751748, 2004588747, 2119892320, 2496984634, 2633593437, 3078393066, 3238866099, 3759275043, 3946321306, 4550691565, 4767170058, 5464303632, 5713223355, ...

Li7_abyme = 1, 2, 6, 9, 10, 15, 20, 23, 28, 31, 32, 35, 75, 76, 91, 92, 107, 108, 123, 124, 264, 279, 284, 299, 304, 319, 324, 339, 344, 359, 364, 379, 384, 399, 404, 419, 424, ...

Mise en abyme des nombres octogonaux (ce lien montre en outre une spirale hexagonale générant tous les hexagonaux) :

8_abyme = 0, 100101, 271201, 620165, 1100496, 1735841, 2060065, 3082560, 3588320, 5137825, 5885601, 8124656, 9184000, 118528816, 150067841, 187878360, 232777825, 285636176, 347375841, 418971736, 501451265, 595894320, 703433281, 825253016, 962590881, ...

K8_abyme = 1, 17, 23, 29, 34, 39, 41, 46, 48, 53, 55, 60, 62, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, ...

Mise en abyme des nombres ennéagonaux :

9_abyme = 0, 75, 325, 10836, 42075, 56134, 344301, 545100, 1449966, 2813889, 5028606, 8404875, 13592016, 19653225, 30368841, 41481264, 60123225, 78624750, ...

Li9_abyme = 1, 2, 3, 8, 12, 13, 22, 25, 33, 40, 47, 54, 62, 69, 78, 85, 94, 101, ...

Mise en abyme des nombres décagonaux :

10_abyme = 0, 85, 9457, 1636480, 3331537, 6166530, 15188557, 23846130, 35967007, 52406740, 74119185, 101914072, 120675717, 137575305, 161270950, 182445802, 211913527, 238061755, 274208760, 306083772, 349886377, 388296877, 440800522, 486610510, 548929755, 603058527, 676377052, 739799200, 825369805, 899115217, 998259822, ...

Li10_abyme = 1, 2, 7, 31, 38, 45, 58, 66, 74, 82, 90, 98, 103, 107, 112, 116, 121, 125, 130, 134, 139, 143, 148, 152, 157, 161, 166, 170, 175, 179, 184, ...

__________

Que pourrait être, sur ce modèle, une mise en abyme des nombres premiers ? Il faut d’abord fabriquer l’« escalier » qui leur correspond.

Comme pour les « escaliers » vus jusqu’ici (où chaque marche se terminait par un nombre triangulaire, carré, pentagonal, etc.), l’« escalier des premiers » voit chacune de ses marches se terminer par un premier. La « forme première » en résulte et le semis peut commencer :

Escalier des premiers Forme première Semis

00 01 02 (3 nombres) * * * (3 cases) 2,3,5 (3 chiffres)

03 (1 nombre) * (1 case) 7 (1 chiffre)

04 05 (2 nombres) * * (2 cases) 1 1 (2 chiffres)

06 07 ... * * ... 1 3

08 09 10 11 * * * * 1 7,1 9

12 13 * * 2 3

14 15 16 17 * * * * 2 9,3 1

18 19 * * 3 7

20 21 22 23 * * * * 4 1,4 3

24 25 26 27 28 29 * * * * * * 4 7,5 3,5 9

30 31 * * 6 1

32 33 34 35 36 37 * * * * * * 6 7,7 1,7 3

38 39 40 41 * * * * 7 9,8 3

42 43 * * 8 9,

44 45 46 47 * * * * 9 7,1 0

48 49 50 51 52 53 * * * * * * 1,1 0 3,1 0

54 55 56 57 58 59 * * * * * * 7,1 0 9,1 1

60 61 * * 3,1

62 63 64 65 66 67 * * * * * * 2 7,1 3 1,1

68 69 70 71 * * * * 3 7,1 3

72 73 * * 9,1

74 75 76 77 78 79 * * * * * * 4 9, 1 5 1,1

80 81 82 83 * * * * 5 7,1 6

84 85 86 87 88 89 * * * * * * 3,1 6 7,1 7

90 91 92 93 94 95 96 97 * * * * * * * * 3,1 8 1,1 9 1,1

... ... ...

La suite Pr_abyme des premiers qui « tombent juste » (même après enjambement) est composée des nombres en jaune ci-dessus ; la suite LiPr_abyme est celle des numéros de ligne où apparaissent un à un ces nombres :

Pr_abyme = 5, 7, 11, 13, 19, 23, 31, 37, 43, 59, 61, 73, 83, 89, 10007, 10037, 10111, 10139, 10163, 10321, 10369, 10399, 10459, 10499, 10513, 10567, 10729, 10831, 10973, 11059, 11113, 11149, 11161, 11279, 11317, 11383, 11467, 11491, 11621, 11657, 11777, 11813, 11863, 11887, 11959, 12041, 12101, 12149, 12263, 12289, 12373, 12511, 12527, 12553, 12619, 12653, 12671, 12713, 12739, 12799, 12917, 13007, 13043, 13063, 13109, 13309, 13463, 13477, 13681, 13697, 13751, 13799, 13873, 13907, 13921, 13967, 13999, 14057, 14087, 14341, 14407, 14447, 14557, 14731, 14753, 14797, 14821, 14869, 14957, 15101, 15173, 15319, 15401, 15427, 15461, 15551, 15607, 15649, 15679, 15739, 15803, 15881, 15919, 15971, 15991, 16087, 16139, 16217, 16349, 16427, 16481, 16699, 16729, 16763, 16811, 16843, 16963, 16981, 17093, 17123, 17293, 17341, 17383, 17389, 17477, 17551, 17599, 17657, 17737, 17791, 17839, 17957, 17989, 18059, 18077, 18131, 18223, 18269, 18379, 18439, 18517, 18583, 18773, 18793, 18899, 18919, 19073, 19081, 19141, 19211, 19219, 19373, 19469, 19477, 19507, 19717, 19739, 19853, 19919, 19997, 20071, 20149, 20393, 20407, 20479, 20543, 20611, 20693, 20731, 20747, 20771, 20857, 20879, 20903, 20929, 21101, 21143, 21157, 21247, 21481, 21557, 21647, 21661, 21713, 21893, 21977, 22013, 22123, 22153, 22159, 22349, 22369, 22409, 22543, 22637, 22691, 22727, 22901, 22963, 22993, 23029, 23057, 23131, 23173, 23357, 23473, 23671, 23753, 23813, 23833, 23893, 23977, 24007, 24023, 24061, 24203, 24281, 24391, 24443, 24499, 24593, 24709, 24749, 24877, 24919, 24977, 25073, 25117, 25183, 25247, 25349, 25411, 25463, 25579, 25657, 25673, 25741, 25771, 25913, 25951, 26029, 26111, 26171, 26203, 26227, 26261, 26317, 26339, 26387, 26431, 26449, 26597, 26647, 26879, 27191, 27283, 27329, 27409, 27539, 27583, 27737, 27767, 27817, 28051, 28069, 28099, 28111, 28181, 28277, 28411, 28517, 28607, 28621, 28649, 28771, 28871, 28901, 28933, 29167, 29327, 29387, 29399, 29429, 29501, 29587, 29633, 29717, 29789, 29819, 29873, 29947, 30029, 30097, 30241, 30367, 30449, 30529, 30559, 30593, 30803, 30839, 30893, 30971, 31079, 31183, 31193, 31249, 31307, 31357, 31393, 31627, 31741, 31849, 31991, 32063, 32117, 32159, 32327, 32353, 32423, 32537, 32587, 32609, 32687, 32839, 32869, 32939, 32987, 33023, 33149, 33199, 33349, 33391, 33479, 33587, 33613, 33641, 33739, 33797, 33829, 33857, 33997, 34157, 34211, 34273, 34303, 34369, 34429, 34487, 34549, 34603, 34679, 34739, 34871, 34913, 35111, 35201, 35317, 35327, 35437, 35527, 35759, 35863, 35983, 36017, 36191, 36353, 36469, 36497, 36677, 36721, 36899, 36913, 37057, 37087, 37139, 37277, 37361, 37493, 37517, 37537, 37607, 37633, 37663, 37781, 37847, 37861, 38167, 38219, 38281, 38317, 38327, 38611, 38713, 38791, 38921, 39113, 39157, 39209, 39323, 39443, 39541, 39619, 39667, 39679, 39749, 39791, 39883, 39901, 39983, 40031, 40387, 40427, 40487, 40519, 40583, 40597, 40763, 40849, 40867, 40903, 40973, 41057, 41177, 41183, 41213, 41257, 41269, 41399, 41603, 41611, 41641, 41729, 41849, 41911, 41999, 42017, 42061, 42131, 42257, 42323, 42407, 42433, 42457, 42677, 42709, 42829, 42859, 42937, 42953, 43237, 43291, 43397, 43597, 43649, 43661, 43789, 43951, 44041, 44059, 44101, 44179, 44269, 44449, 44483, 44531, 44641, 44699, 44819, 44887, 44983, 45137, 45161, 45289, 45307, 45337, 45403, 45427, 45491, 45569, 45757, 45767, 45823, 45893, 46133, 46171, 46229, 46351, 46447, 46511, 46589, 46861, 47059, 47279, 47303, 47317, 47363, 47387, 47501, 47513, 47659, 47717, 47791, 47969, 48079, 48409, 48589, 48649, 48673, 48733, 48757, 48799, 48989, 49003, 49033, 49117, 49157, 49211, 49277, 49477, 49537, 49597, 49669, 49727, 49741, 49787, 49927, 49957, 50273, 50359, 50417, 50503, 50627, 50789, 50867, 50993, 51133, 51197, 51241, 51487, 51521, 51551, 51673, 51713, 51803, 51971, 51977, 52027, 52177, 52183, 52291, 52387, 52501, 52579, 52667, 52691, 52813, 52951, 52973, 53051, 53077, 53117, 53161, 53173, 53239, 53309, 53381, 53593, 53629, 53639, 53693, 53717, 53857, 53939, 54011, 54091, 54181, 54287, 54323, 54421, 54497, 54673, 54751, 54973, 55103, 55229, 55343, 55373, 55439, 55457, 55619, 55663, 55691, 55763, 55817, 55823, 55921, 56003, 56113, 56179, 56249, 56369, 56543, 56629, 57203, 57349, 57389, 57559, 57587, 57641, 57653, 57697, 57713, 57737, 57853, 57881, 58013, 58031, 58073, 58379, 58441, 58481, 58733, 59009, 59021, 59069, 59119, 59183, 59219, 59359, 59407, 59471, 59513, 59621, 59659, 59707, 59791, 59863, 59957, 60139, 60353, 60383, 60457, 60623, 60661, 60719, 60887, 61001, 61057, 61151, 61223, 61333, 61363, 61381, 61441, 61469, 61631, 61729, 61837, 61879, 61981, 62039, 62137, 62201, 62297, 62303, 62581, 62633, 62861, 62873, 62927, 63079, 63127, 63211, 63317, 63353, 63367, 63521, 63671, 63781, 63853, 64007, 64153, 64171, 64301, 64319, 64403, 64567, 64627, 64871, 64921, 64969, 65033, 65147, 65203, 65287, 65309, 65381, 65437, 65599, 65707, 65963, 66041, 66103, 66109, 66337, 66361, 66499, 66643, 66841, 66853, 67033, 67049, 67121, 67217, 67261, 67411, 67453, 67481, 67511, 67531, 67567, 67741, 67763, 67819, 67843, 67943, 67961, 67993, 68161, 68227, 68329, 68371, 68449, 68501, 68539, 68659, 68699, 68863, 68897, 68947, 68993, 69197, 69317, 69341, 69389, 69403, 69467, 69493, 69499, 69661, 69859, 69929, 70177, 70183, 70289, 70309, 70487, 70501, 70573, 70627, 70769, 70793, 70951, 70969, 71089, 71293, 71341, 71389, 71633, 71843, 71947, 71999, 72089, 72109, 72161, 72211, 72223, 72431, 72643, 72679, 72733, 72869, 72893, 72907, 73013, 73043, 73063, 73277, 73459, 73553, 73607, 73699, 73859, 73939, 74017, 74161, 74521, 74573, 74597, 74699, 74731, 74941, 75011, 75079, 75149, 75167, 75503, 75577, 75703, 75821, 75853, 75941, 75989, 75997, 76039, 76207, 76231, 76649, 76717, 76771, 76831, 76847, 76907, 76919, 77041, 77153, 77201, 77261, 77339, 77369, 77509, 77521, 77563, 77617, 77761, 77863, 78059, 78101, 78157, 78193, 78277, 78301, 78571, 78803, 78853, 78877, 78901, 79039, 79153, 79283, 79367, 79399, 79423, 79621, 79769, 79801, 79823, 79867, 80077, 80153, 80231, 80489, 80567, 80681, 80747, 80789, 80897, 80933, 81101, 81181, 81239, 81373, 81551, 81727, 81853, 81883, 81971, 82037, 82073, 82139, 82231, 82267, 82351, 82469, 82507, 82559, 82567, 82609, 82723, 82781, 82811, 82997, 83009, 83063, 83117, 83273, 83423, 83449, 83471, 83557, 83609, 83621, 83777, 83933, 84089, 84221, 84401, 84449, 84737, 84793, 84979, 85049, 85103, 85223, 85297, 85363, 85439, 85451, 85523, 85577, 85643, 85711, 85847, 86111, 86117, 86209, 86243, 86323, 86389, 86677, 86851, 86969, 86993, 87071, 87133, 87181, 87253, 87323, 87403, 87473, 87629, 87679, 87743, 87797, 87811, 87877, 87887, 87943, 88117, 88379, 88469, 88493, 88661, 88741, 88817, 88843, 89293, 89381, 89413, 89477, 89501, 89597, 89653, 89689, 89759, 89797, 89849, 89891, 90071, 90149, 90263, 90281, 90373, 90481, 90533, 90631, 90647, 90697, 90709, 90887, 91141, 91229, 91253, 91369, 91411, 91457, 91591, 91631, 91807, 91837, 91867, 91969, 92003, 92143, 92203, 92221, 92269, 92431, 92581, 92647, 92801, 93113, 93179, 93257, 93287, 93559, 93637, 93701, 93887, 93979, 94009, 94049, 94099, 94111, 94321, 94447, 94613, 94781, 94793, 94837, 95063, 95191, 95239, 95339, 95383, 95419, 95539, 95561, 95701, 95813, 95873, 96013, 96167, 96289, 96323, 96353, 96451, 96517, 96589, 96959, 97003, 97103, 97213, 97381, 97397, 97453, 97583, 97649, 97673, 97771, 97871, 98011, 98297, 98411, 98429, 98467, 98479, 98561, 98689, 98713, 98947, 98993, 99013, 99083, 99119, 99173, 99397, 99971, ...

LiPr_abyme = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 637, 639, 642, 645, 648, 655, 659, 661, 666, 668, 670, 674, 684, 687, 696, 699, 703, 705, 706, 711, 714, 720, 725, 727, 734, 736, 739, 744, 748, 750, 756, 759, 763, 767, 777, 778, 781, 788, 790, 792, 797, 800, 801, 803, 804, 807, 814, 823, 826, 827, 831, 840, 847, 848, 857, 860, 865, 868, 871, 875, 876, 877, 880, 883, 886, 898, 902, 906, 910, 919, 922, 926, 928, 929, 935, 939, 942, 953, 961, 963, 966, 970, 973, 978, 980, 983, 987, 989, 992, 994, 997, 1002, 1006, 1007, 1012, 1018, 1022, 1033, 1035, 1036, 1038, 1041, 1046, 1047, 1053, 1056, 1061, 1066, 1067, 1068, 1073, 1078, 1083, 1086, 1092, 1095, 1096, 1105, 1108, 1109, 1111, 1114, 1121, 1124, 1129, 1132, 1137, 1141, 1150, 1151, 1154, 1157, 1163, 1164, 1168, 1172, 1174, 1180, 1186, 1188, 1191, 1200, 1202, 1209, 1213, 1219, 1223, 1229, 1239, 1241, 1243, 1248, 1252, 1254, 1255, 1257, 1261, 1264, 1265, 1268, 1270, 1277, 1278, 1281, 1286, 1295, 1301, 1307, 1309, 1311, 1319, 1323, 1326, 1332, 1335, 1336, 1343, 1345, 1346, 1354, 1357, 1359, 1363, 1372, 1374, 1375, 1378, 1380, 1384, 1388, 1395, 1399, 1409, 1412, 1416, 1420, 1422, 1426, 1428, 1430, 1433, 1443, 1446, 1451, 1453, 1457, 1459, 1465, 1466, 1472, 1474, 1476, 1482, 1484, 1488, 1493, 1500, 1503, 1507, 1512, 1517, 1518, 1521, 1523, 1529, 1532, 1534, 1540, 1542, 1545, 1547, 1549, 1552, 1554, 1556, 1557, 1559, 1566, 1569, 1581, 1596, 1601, 1602, 1604, 1609, 1612, 1619, 1624, 1629, 1638, 1641, 1643, 1645, 1648, 1649, 1656, 1661, 1664, 1666, 1668, 1673, 1677, 1678, 1680, 1691, 1699, 1701, 1703, 1708, 1710, 1714, 1716, 1719, 1721, 1723, 1728, 1733, 1735, 1739, 1747, 1754, 1755, 1758, 1760, 1762, 1772, 1774, 1778, 1781, 1789, 1796, 1798, 1801, 1804, 1808, 1809, 1818, 1824, 1831, 1834, 1840, 1844, 1847, 1855, 1857, 1861, 1865, 1868, 1871, 1875, 1880, 1883, 1885, 1889, 1891, 1894, 1898, 1905, 1908, 1912, 1916, 1919, 1920, 1924, 1929, 1931, 1933, 1938, 1941, 1944, 1947, 1950, 1955, 1956, 1960, 1965, 1968, 1970, 1974, 1978, 1981, 1988, 1993, 1998, 2001, 2006, 2011, 2020, 2026, 2032, 2034, 2040, 2048, 2051, 2055, 2065, 2067, 2079, 2080, 2088, 2090, 2092, 2100, 2103, 2109, 2111, 2113, 2118, 2119, 2122, 2125, 2126, 2128, 2136, 2139, 2142, 2144, 2146, 2152, 2157, 2161, 2166, 2174, 2176, 2179, 2186, 2191, 2192, 2197, 2200, 2203, 2207, 2211, 2217, 2218, 2222, 2224, 2236, 2238, 2240, 2242, 2245, 2246, 2251, 2257, 2258, 2261, 2265, 2269, 2274, 2276, 2278, 2282, 2284, 2288, 2294, 2296, 2299, 2304, 2310, 2313, 2317, 2318, 2321, 2322, 2328, 2330, 2335, 2338, 2341, 2347, 2352, 2358, 2360, 2363, 2366, 2376, 2379, 2384, 2391, 2392, 2395, 2400, 2406, 2411, 2413, 2416, 2420, 2428, 2433, 2434, 2436, 2443, 2446, 2453, 2456, 2460, 2465, 2466, 2472, 2474, 2477, 2480, 2482, 2484, 2489, 2494, 2497, 2499, 2501, 2509, 2512, 2515, 2520, 2524, 2525, 2530, 2546, 2551, 2560, 2562, 2563, 2564, 2565, 2573, 2576, 2583, 2586, 2590, 2597, 2602, 2612, 2621, 2624, 2626, 2628, 2631, 2632, 2640, 2642, 2645, 2648, 2652, 2655, 2658, 2668, 2672, 2674, 2679, 2682, 2685, 2687, 2694, 2695, 2706, 2710, 2714, 2716, 2722, 2726, 2728, 2733, 2737, 2739, 2742, 2755, 2756, 2757, 2763, 2765, 2768, 2776, 2777, 2780, 2785, 2787, 2790, 2795, 2799, 2803, 2806, 2808, 2811, 2817, 2820, 2823, 2825, 2828, 2830, 2831, 2833, 2837, 2839, 2847, 2849, 2850, 2853, 2855, 2861, 2866, 2867, 2871, 2874, 2878, 2880, 2887, 2889, 2899, 2903, 2912, 2916, 2920, 2925, 2927, 2931, 2933, 2939, 2941, 2943, 2947, 2951, 2952, 2957, 2961, 2965, 2969, 2972, 2976, 2983, 2988, 3012, 3017, 3018, 3023, 3026, 3029, 3031, 3034, 3036, 3039, 3043, 3044, 3049, 3051, 3054, 3069, 3071, 3073, 3080, 3089, 3090, 3094, 3097, 3099, 3101, 3109, 3113, 3119, 3123, 3127, 3129, 3132, 3136, 3138, 3141, 3150, 3157, 3159, 3161, 3164, 3169, 3171, 3179, 3184, 3187, 3190, 3191, 3197, 3200, 3201, 3202, 3203, 3210, 3216, 3218, 3221, 3223, 3225, 3229, 3231, 3234, 3236, 3244, 3247, 3255, 3257, 3259, 3267, 3269, 3271, 3275, 3277, 3279, 3285, 3292, 3298, 3302, 3305, 3312, 3314, 3317, 3318, 3320, 3327, 3332, 3340, 3341, 3343, 3346, 3352, 3356, 3360, 3362, 3364, 3369, 3378, 3384, 3391, 3395, 3398, 3399, 3403, 3406, 3411, 3418, 3424, 3425, 3429, 3430, 3433, 3440, 3443, 3449, 3451, 3452, 3455, 3456, 3457, 3464, 3467, 3470, 3472, 3479, 3480, 3483, 3488, 3491, 3494, 3496, 3499, 3503, 3505, 3507, 3509, 3515, 3519, 3521, 3523, 3529, 3535, 3537, 3539, 3540, 3544, 3546, 3547, 3550, 3559, 3561, 3570, 3573, 3577, 3579, 3586, 3587, 3590, 3592, 3600, 3603, 3609, 3610, 3618, 3627, 3630, 3632, 3641, 3645, 3653, 3655, 3658, 3662, 3663, 3666, 3667, 3676, 3683, 3688, 3691, 3693, 3696, 3698, 3703, 3706, 3707, 3714, 3719, 3724, 3726, 3731, 3733, 3736, 3740, 3749, 3763, 3766, 3767, 3769, 3773, 3784, 3786, 3788, 3790, 3792, 3799, 3804, 3810, 3816, 3817, 3819, 3822, 3823, 3827, 3832, 3833, 3848, 3851, 3854, 3857, 3858, 3859, 3860, 3862, 3867, 3869, 3871, 3877, 3879, 3885, 3886, 3889, 3893, 3903, 3906, 3914, 3916, 3919, 3921, 3924, 3925, 3935, 3944, 3945, 3946, 3947, 3951, 3957, 3962, 3965, 3966, 3968, 3980, 3986, 3987, 3990, 3992, 3997, 4000, 4003, 4012, 4016, 4021, 4023, 4024, 4027, 4030, 4039, 4043, 4048, 4052, 4056, 4060, 4065, 4066, 4072, 4076, 4080, 4081, 4087, 4089, 4091, 4096, 4098, 4101, 4102, 4103, 4107, 4109, 4111, 4116, 4118, 4120, 4124, 4132, 4136, 4138, 4140, 4142, 4146, 4147, 4152, 4156, 4162, 4167, 4173, 4177, 4191, 4194, 4199, 4202, 4205, 4209, 4211, 4214, 4218, 4219, 4223, 4224, 4228, 4230, 4232, 4237, 4240, 4243, 4245, 4251, 4254, 4261, 4268, 4272, 4274, 4279, 4282, 4284, 4286, 4290, 4291, 4294, 4301, 4305, 4310, 4313, 4314, 4318, 4319, 4321, 4325, 4334, 4335, 4336, 4341, 4345, 4348, 4350, 4372, 4373, 4376, 4379, 4381, 4386, 4391, 4393, 4396, 4397, 4400, 4401, 4411, 4415, 4420, 4421, 4424, 4429, 4433, 4436, 4438, 4442, 4444, 4449, 4457, 4461, 4465, 4470, 4473, 4475, 4481, 4483, 4486, 4489, 4490, 4495, 4496, 4501, 4503, 4504, 4506, 4513, 4517, 4519, 4526, 4537, 4538, 4545, 4547, 4556, 4560, 4563, 4565, 4569, 4570, 4571, 4574, 4576, 4585, 4593, 4601, 4606, 4609, 4612, 4617, 4621, 4622, 4629, 4631, 4633, 4638, 4639, 4644, 4651, 4653, 4660, 4665, 4670, 4672, 4674, 4677, 4678, 4681, 4693, 4696, 4699, 4706, 4710, 4711, 4713, 4718, 4719, 4720, 4722, 4728, 4733, 4741, 4746, 4748, 4750, 4751, 4754, 4755, 4757, 4767, 4769, 4771, 4772, 4775, 4778, 4786, 4809, ...

Une mise en abyme équivalente des nombres de Fibonacci ne produit pas grand chose, comme l’a vérifié Jean-Marc Falcoz : « Jusqu'à F30000, les seuls nombres qui marchent sont 1, 1, 2, 5, 13 » :

Fi_abyme = 1, 1, 2, 5, 13, ...

LiFi_abyme = 1, 2, 3, 4, 5, ...

Merci à tous !

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