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%I #25 Oct 29 2022 04:51:14
%S 1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,2,1,2,
%T 1,3,1,2,1,2,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,3,1,2,1,3,
%U 1,2,1,2,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,2,1,2,1,3,1,2,1,2,1
%N Sequence is S(infinity), where S(1) = 1, S(m+1) = concatenation S(m), a(m)+1, S(m) and a(m) is the m-th term of S(m). a(m) is also the m-th term of the sequence.
%C S(m) has 2^m - 1 elements and is palindromic for all m.
%C First occurrence of k: 1,2,4,16,65536,...,. A014221: a(n+1) = 2^a(n). This is an Ackermann function. - _Robert G. Wilson v_, May 30 2006
%H Robert G. Wilson v, <a href="/A089242/b089242.txt">Table of n, a(n) for n = 1..65536</a>
%F a(m) = number of c's such that 0 = c(c(c(c(...c(m)...)))), where 2^c(n) is the highest power of 2 which divides evenly into n (i.e., a(m) = 1 + a(c(m))); also c(m) = A007814(m).
%F In other words, a(n) = number of iterates of A007814 until a zero is encountered.
%F Multiplicative with a(2^e) = 1 + a(e), a(p^e) = 1 for odd prime p. - _Andrew Howroyd_, Jul 27 2018
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} a(2^(k-1))/2^k = 1.6980544744753405... . - _Amiram Eldar_, Oct 29 2022
%t c[n_] := (i++; Block[{k = 0, m = n}, While[ EvenQ[m], k++; m /= 2]; k]); f[n_] := (i = 0; NestWhile[c, n, # >= 1 &]; i); Array[f, 105] (* _Robert G. Wilson v_, May 30 2006 *)
%Y Cf. A007814, A014221.
%K nonn,easy,mult
%O 1,2
%A _Leroy Quet_, Dec 13 2003
%E More terms from _David Wasserman_, Aug 31 2005
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