OFFSET
0,2
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.
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..130 [Received Aug 30, 2009]
EXAMPLE
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.
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:
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Yasutoshi Kohmoto, May 30 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Jun 21 2009
Edited by Franklin T. Adams-Watters, Jun 22 2009
STATUS
approved