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

1,2

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

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

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 *)

Crossrefs

Cf. A007814, A014221.

Sequence in context: A327533 A327518 A174532 * A349258 A349326 A185894

Adjacent sequences: A089239 A089240 A089241 * A089243 A089244 A089245

Keyword

nonn,easy,mult

Author

Leroy Quet, Dec 13 2003

Extensions

More terms from David Wasserman, Aug 31 2005

Status

approved