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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 August 4 13:26 EDT 2024. Contains 374921 sequences. (Running on oeis4.)