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A151659
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Terminal point of the repeated application of usigma starting at 2^n.
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3
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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; internal format)
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OFFSET
| 0,2
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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.
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LINKS
| R. J. Mathar, Table of n, a(n) for n=0..130 [Received Aug 30, 2009]
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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.
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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
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CROSSREFS
| Cf. A146892,A161946,A000265,A034448,A006519.
Sequence in context: A000195 A135663 A090620 * A057467 A074594 A001069
Adjacent sequences: A151656 A151657 A151658 * A151660 A151661 A151662
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KEYWORD
| nonn
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AUTHOR
| Yasutoshi Kohmoto (zbi74583.boat at orange.zero.jp), May 30 2009
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EXTENSIONS
| 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
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