cp's OEIS Frontend

A151659 Terminal point of the repeated application of usigma starting at 2^n.

%I A151659 #18 Feb 02 2025 08:48:48
%S A151659 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,
%T A151659 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,
%U A151659 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
%N A151659 Terminal point of the repeated application of usigma starting at 2^n.
%C A151659 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.
%C A151659 The sequence is defined by a(n)=1/c(k), the inverse of the auxiliary sequence c at the point where b terminates.
%C A151659 All values of the sequence are powers of 2.
%H A151659 R. J. Mathar, <a href="/A151659/b151659.txt">Table of n, a(n) for n = 0..130</a> [Received Aug 30, 2009]
%e A151659 The irregular table of the sequences b(.) is in row n=0,1,2,... represented by
%e A151659     1;
%e A151659     2,   3,   1;
%e A151659     4,   5,   3,   1;
%e A151659     8,   9,   5,   3,   1;
%e A151659    16,  17,   9,   5,   3,   1;
%e A151659    32,  33,   3,   1;
%e A151659    64,  65,  21,   1;
%e A151659   128, 129,  11,   3,   1;
%e A151659 The associated table of the sequences c(.) in row n=0,1,2,... is
%e A151659    1;
%e A151659    2,   2,  1/2;
%e A151659    4,   4,   2,  1/2;
%e A151659    8,   8,   4,   2,  1/2;
%e A151659   16,  16,   8,   4,   2,  1/2;
%e A151659   32,  32,   2,  1/2;
%e A151659   64,  64,  16,  1/2;
%e A151659 The reciprocals of the final entries in the rows give the sequence.
%p A151659 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:
%p A151659 A000265 := proc(n,p) option remember; local nshf ; nshf := n ; while (nshf mod p ) = 0 do nshf := nshf/p ; od: nshf ; end:
%p A151659 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:
%p A151659 A161946 := proc(n) option remember; A000265(A034448(n),2) ; end:
%p A151659 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:
%p A151659 seq(A151659(n),n=0..130) ; # _R. J. Mathar_, Aug 31 2009
%Y A151659 Cf. A146892, A161946, A000265, A034448, A006519.
%K A151659 nonn
%O A151659 0,2
%A A151659 _Yasutoshi Kohmoto_, May 30 2009
%E A151659 Edited and extended by _R. J. Mathar_, Jun 21 2009
%E A151659 Edited by _Franklin T. Adams-Watters_, Jun 22 2009