Methodically filling
successive squares and primes with N
[June
16th, 2010]
Hello
SeqFans,
We
want the successive squares to be replaced by sums in this way:
n = 1 2
3 4 5
6 7 8
9 10 11
12 ...
n^2=
1 4
9 16 25
36 49 64
81 100 121
144
-
- - --
-- -- --
-- -- ---
--- ---
Sum 1 4 2
3 6 11 14 16
18 22 26
30
+
+ + +
+ + +
+ + +
7
5 9 12
15 17 19
24 27 32
+ +
+ + +
+ + + +
8 10
13 20 31
21 25 28
33
+ +
+ +
23 29
40 49
Thus,
to complete a sum, we always use the smallest available positive integer not
used so far and not leading to a contradiction.
The
sequence arising from this could be (this is a permutation of N):
Sq =
1,4,2,7,3,5,8,6,9,10,11,12,13,14,15,20,16,17,31,18,19,21,23, ...
The
same algorithm can be used to "fill the Primes":
2 3
5 7 11 13 17
19 23 29
31 ...
- -
- - --
-- -- --
-- -- --
2 3
1 7 5
13 8 19
11 14 10
+ + + +
+ +
4 6 9 12
15 21
... producing:
Pr =
2,3,1,4,7,5,6,13,8,9,19,11,12,14,15,10,21, ...
Neither
"Sq" nor
"Pr" are in the OEIS. If someone could check, extend those seq and submit them (in September -- if they are of
interest), it would be nice.
Best,
É.
_______________
[Douglas McNeil, a couple of hours later]
Can’t
submit it myself, but FYI (and hopefully for comparison with someone else’s
values):
sage: Sq
[1,
4, 2, 7, 3, 5, 8, 6, 9, 10, 11, 12, 13, 14, 15, 20, 16, 17, 31, 18, 19, 21, 23,
22, 24, 25, 29, 26, 27, 28, 40, 30, 32, 33, 49, 34, 35, 36, 64, 37, 38, 39, 82,
41, 42, 43, 44, 55, 45, 46, 47, 48, 70, 50, 51, 52, 53, 83, 54, 56, 57, 58, 99,
59, 60, 61, 62, 119, 63, 65, 66, 67, 68, 71, 69, 72, 73, 74, 75, 78, 76, 77,
79, 80, 81, 91, 84, 85, 86, 87, 89, 98, 88, 90, 92, 93, 95, 118, 94, 96, 97,
100, 101, 137, 102, 103, 104, 105, 106, 156, 107, 108, 109, 110, 111, 184, 112,
113, 114, 115, 116, 214, 117, 120, 121, 122, 123, 238, 124, 125, 126, 127, 128,
129, 141, 130, 131, 132, 133, 134, 135, 166, 136, 138, 139, 140, 142, 143, 186,
144, 145, 146, 147, 148, 149, 210, 150, 151, 152, 153, 154, 155, 241, 157, 158,
159, 160, 161, 162, 268, 163, 164, 165, 167, 168, 169, 300, 170, 171, 172, 173,
174, 175, 334, 176, 177, 178, 179, 180, 181, 182, 191, 183, 185, 187, 188, 189,
190, 192, 207, 193, 194, 195, 196, 197, 198, 199, 228, 200, 201, 202, 203, 204,
205, 206, 260, 208, 209, 211, 212, 213, 215, 216, 280, 217, 218, 219, 220, 221,
222, 223, 309, 224, 225, 226, 227, 229, 230, 231, 344, 232, 233, 234, 235, 236,
237, 239, 379, 240, 242, 243, 244, 245, 246, 247, 409, 248, 249, 250, 251, 252,
253, 254, 452, 255, 256, 257, 258, 259, 261, 262, 496, 263, 264, 265, 266, 267,
269, 270, 537, 271, 272, 273, 274, 275, 276, 277, 278, 304]
sage: Pr
[2,
3, 1, 4, 7, 5, 6, 13, 8, 9, 19, 11, 12, 14, 15, 10, 21, 17, 20, 16, 25, 43, 18,
29, 22, 31, 23, 36, 24, 37, 26, 41, 27, 44, 28, 45, 30, 49, 32, 51, 33, 56, 34,
63, 35, 66, 38, 65, 39, 68, 40, 69, 42, 71, 46, 81, 47, 84, 48, 89, 52, 87, 50,
99, 53, 98, 54, 103, 55, 108, 57, 110, 58, 115, 59, 120, 60, 121, 61, 130, 62,
64, 67, 70, 127, 73, 126, 72, 139, 74, 149, 75, 152, 76, 153, 77, 156, 78, 79,
82, 80, 161, 83, 168, 85, 172, 86, 177, 88, 90, 91, 92, 179, 93, 184, 94, 187,
95, 188, 96, 97, 100, 101, 102, 104, 105, 206, 106, 207, 107, 210, 109, 222,
111, 112, 114, 113, 116, 118, 117, 232, 119, 234, 122, 237, 123, 244, 124, 249,
125, 254, 128, 255, 129, 260, 131, 132, 134, 133, 268, 135, 136, 138, 137, 140,
142, 141, 280, 143, 288, 144, 289, 145, 146, 148, 147, 296, 150, 299, 151, 306,
154, 307, 155, 308, 157, 310, 158, 159, 162, 160, 163, 164, 165, 326, 166, 333,
167, 336, 169, 340, 170, 171, 180, 173, 174, 176, 175, 181, 185, 178, 183, 186,
182, 375, 189, 374, 190, 379, 191, 380, 192, 385, 193, 194, 200, 195, 196, 202,
197, 198, 204, 199, 402, 201, 406, 203, 410, 205, 412, 208, 411, 209, 422, 211,
212, 218, 213, 214, 216, 215, 432, 217, 436, 219, 440, 220, 441, 221, 223, 229,
224, 225, 228, 226, 227, 230, 231, 460, 233, 468, 235, 236, 238, 239, 480, 240,
241, 246, 242, 243, 248, 245, 494, 247, 496, 250, 501, 251, 506, 252, 253, 256,
257, 512, 258, 515, 259, 261, 267, 262, 263, 272, 264, 266, 279, 265, 269, 277,
270, 273, 278, 271, 552, 274, 553, 275, 554, 276, 281, 282, 283, 284, 286, 285,
572, 290, 569, 287, 576, 291, 292, 294, 293, 588, 297, 586, 295, 592, 298, 300,
309, 301, 610, 302, 303, 314, 304, 305, 320, 311, 626, 312, 313, 316, 315, 632,
317, 636, 318, 319, 330, 321, 322, 328, 323, 325, 329, 324, 327, 332, 331, 660,
334, 663]
Your
sequences are always fun to program!
Doug
_______________
[Me, a couple of seconds later]
Again,
many, many thanks, Doug! And yes,
please submit and co-author in September (...)!
Best,
É.