A089242 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.

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, 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, 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

Offset: 1

Author

Leroy Quet, Dec 13 2003

Keywords

• nonn
• easy
• mult

Comments

S(m) has 2^m - 1 elements and is palindromic for all m.

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

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..65536

Crossrefs

Cf. A007814, A014221.

Programs

Mathematica

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 *)

Formula

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).

In other words, a(n) = number of iterates of A007814 until a zero is encountered.

Multiplicative with a(2^e) = 1 + a(e), a(p^e) = 1 for odd prime p. - Andrew Howroyd, Jul 27 2018

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

Extensions

More terms from David Wasserman, Aug 31 2005