A151659 Terminal point of the repeated application of usigma starting at 2^n.

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 2, 4, 4, 8, 4, 8, 8, 4, 4, 4, 4, 8, 8, 4, 8, 4, 8, 4, 8, 8, 16, 8, 8, 16, 4, 8, 16, 8, 32, 16, 8, 8, 8, 8, 32, 8, 16, 8, 32, 16, 32, 8, 16, 16, 16, 32, 16, 16, 16, 8, 16, 16, 16, 16, 16, 8, 16, 16, 8, 16, 16, 64, 8, 32, 32, 16

Offset: 0

Author

Yasutoshi Kohmoto, May 30 2009

Keywords

• nonn

Comments

For each n, we define an auxiliary sequence b(k) starting at b(0)=2^n by b(k+1) = A161946( b(k) ) = A000265(A034448( b(k) )), that is, repeated application of the unitary sigma value to its odd part. b(k) terminates at some k with b(k)=1. In addition there is an auxiliary parallel sequence c(k) defined by c(0)=2^n and recursively c(k+1) = c(k)/A006519(A034448(b(k))), reducing 2^n by the powers of 2 which are divided out of the sequence b.

The sequence is defined by a(n)=1/c(k), the inverse of the auxiliary sequence c at the point where b terminates.

All values of the sequence are powers of 2.

Examples

			The irregular table of the sequences b(.) is in row n=0,1,2,... represented by
    1;
    2,   3,   1;
    4,   5,   3,   1;
    8,   9,   5,   3,   1;
   16,  17,   9,   5,   3,   1;
   32,  33,   3,   1;
   64,  65,  21,   1;
  128, 129,  11,   3,   1;
The associated table of the sequences c(.) in row n=0,1,2,... is
   1;
   2,   2,  1/2;
   4,   4,   2,  1/2;
   8,   8,   4,   2,  1/2;
  16,  16,   8,   4,   2,  1/2;
  32,  32,   2,  1/2;
  64,  64,  16,  1/2;
The reciprocals of the final entries in the rows give the sequence.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 0..130 [Received Aug 30, 2009]

Crossrefs

Cf. A146892, A161946, A000265, A034448, A006519.

Programs

Maple

A034448 := proc(n) local ans, i: ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: ans ; end:
A000265 := proc(n,p) option remember; local nshf ; nshf := n ; while (nshf mod p ) = 0 do nshf := nshf/p ; od: nshf ; end:
A006519 := proc(n) local nshf,a ; a := 1; nshf := n ; while (nshf mod 2 ) = 0 do nshf := nshf/2 ; a := a*2 ; od: a ; end:
A161946 := proc(n) option remember; A000265(A034448(n),2) ; end:
A151659 := proc(n) local b,a ; b := [2^n] ; while op(-1,b) <> 1 do b := [op(b), A161946(op(-1,b)) ] ; od: a := 2^n ; for k from 2 to nops(b) do a := a/ A006519(A034448(op(k-1,b))) ; od: 1/a ; end:
seq(A151659(n),n=0..130) ; # R. J. Mathar, Aug 31 2009

Extensions

Edited and extended by R. J. Mathar, Jun 21 2009

Edited by Franklin T. Adams-Watters, Jun 22 2009