

A151659


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


3



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
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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:
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(k1, b))) ; od: 1/a ; end:
seq(A151659(n), n=0..130) ; # R. J. Mathar, Aug 31 2009


CROSSREFS

Cf. A146892,A161946,A000265,A034448,A006519.
Sequence in context: A000195 A135663 A090620 * A057467 A074594 A211667
Adjacent sequences: A151656 A151657 A151658 * A151660 A151661 A151662


KEYWORD

nonn


AUTHOR

Yasutoshi Kohmoto, May 30 2009


EXTENSIONS

Edited and extended by R. J. Mathar, Jun 21 2009
Edited by Franklin T. AdamsWatters, Jun 22 2009


STATUS

approved



