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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 (list; graph; refs; listen; history; text; internal format)
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(k-1, 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. Adams-Watters, Jun 22 2009

STATUS

approved

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Last modified July 12 02:02 EDT 2020. Contains 335658 sequences. (Running on oeis4.)