f(a(n)) = a(a(n))

On nov. 2^nd, 2009 I have sent this msg to the < Seqfan Mailing list - http://list.seqfan.eu/ >

Hello SeqFans,

If we have 2.a(n) = a(a(n)) then, I guess, the resulting sequence S starts like this:

S = 2,4,1,8,6,12,9,16,18,11,22,24,14,28,17,32,34,36,20,40,23,44,46,48,26,52,29,56,...

(always take the smallest a(n) not yet present and not leading to a contradiction)

Best,

É.

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Jean-Marc Falcoz was quick to send me this mail (in French):

Salut Eric,

Ces jours, je suis tombé par hasard sur une de tes trouvailles :

ces suites où tu demandes par exemple que 2 a(n) =a(a(n)), ou plus généralement, que f(a(n)) = a(a(n))

J’ai écrit un petit programme qui construit la suite quand on donne une fonction f quelconque [dans ton exemple, on aurait eu f(x) = 2x]

Je l’ai essayé avec différentes f :

f(x) = 2x

f(x) = 13x

f(x) = 3x + 1

f(x) = 3x^2 - 1

f(x) = x^ème nombre premier

Voici ce que ça donne numériquement et graphiquement : http://vargo.chez.com/eric.htm

(...)

(if Jean-Marc’s link is not responding, see the content of his page below -- I’m impressed!)

I reproduce here the opening sequence (and the nice zigzag graph) computed by Jean-Marc for the first 49 integers:

S = 2, 4, 1, 8, 6, 12, 9, 16, 18, 11, 22, 24, 14, 28, 17, 32, 34, 36, 20, 40, 23, 44, 46, 48, 26, 52, 29, 56, 58, 31, 62, 64, 35, 68, 70, 72, 38, 76, 41, 80, 82, 43, 86, 88, 47, 92, 945, 96, 50, ...

Merci beaucoup, Jean-Marc !

Best,

É.

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More sequences?

< http://www.cetteadressecomportecinquantesignes.com/Sweet.htm >