0,2

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.

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

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.

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:

read("transforms3") ; L := [seq(A151659(n), n=0..130)] ; LISTTOBFILE("b151659.txt", L, 0) ; # R. J. Mathar, Aug 31 200

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

Sequence in context: A000195 A135663 A090620 this_sequence A057467 A074594 A001069

Adjacent sequences: A151656 A151657 A151658 this_sequence A151660 A151661 A151662

nonn

Yasutoshi Kohmoto (zbi74583.boat at orange.zero.jp), May 30 2009

Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 21 2009

Edited by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 22 2009