A146892 -id:A146892 - cp's OEIS frontend

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

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

Offset: 0

Yasutoshi Kohmoto, May 30 2009

nonn

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.

			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.

R. J. Mathar, Table of n, a(n) for n = 0..130 [Received Aug 30, 2009]

Cf. A146892, A161946, A000265, A034448, A006519.

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

Edited and extended by R. J. Mathar, Jun 21 2009

Edited by Franklin T. Adams-Watters, Jun 22 2009

A146891 Terminal point of a repeated reduction of usigma starting at 2^n.

Original entry on oeis.org

1, 6, 20, 72, 72, 72, 20, 72, 72, 17280, 4800, 17280, 72, 17280, 1152000, 5184, 5184, 5184, 96000, 5184, 345600, 1244160, 320000, 1244160, 82944000, 89579520, 71663616000, 298598400, 1244160, 82944000, 23040000, 82944000, 19906560000

Offset: 0

Yasutoshi Kohmoto, Apr 17 2009

nonn

Let PF_p(n) be the highest power of p dividing n. Examples are PF_2(n) = A006519(n), PF_3(n) = A038500(n) and PF_5(n) = 5^A112765(n) for the cases p = 2, 3, and 5.
Multi-indexed PF_(p1,p2,...)(n) are defined as the products PF_(p1)(n)*PF_(p2)(n)*...
For each n, we define an auxiliary sequence b(k) starting at b(0) = 2^n by b(k+1) = A034448(b(k))/PF_(2,3,5)(A034448(b(k)), that is, repeated removal of all powers of 2, 3 and 5 from the unitary sigma value. 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)*PF_(3,5)(A034448(b(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) = c(k), the auxiliary sequence c at the point where b terminates.
All values of the sequence a(n) are 5-smooth, i.e., members of A051037.

			n=5
b(n) : 2^5 -> 11 -> 1
c(n) : 2^5 -> 2^5*3 -> 2^3*3^2
So a(5) = c(2) = 2^3*3^2 = 72.

Cf. A146892, A151659.

PF := proc(n,p) local nshf,a ; a := 1; nshf := n ; while (nshf mod p ) = 0 do nshf := nshf/p ; a := a*p ; od: a ; end:
A146891 := proc(n) local b,a,k,t ;
    b := [2^n] ;
    while op(-1,b) <> 1 do
        t := A034448(op(-1,b)) ;
        b := [op(b), t/A006519(t)/ A038500(t)/PF(t,5) ] ;
    od:
    a := 2^n ;
    for k from 2 to nops(b) do
        t := A034448(op(k-1,b)) ;
        a := a/ A006519(t) *A038500(t)*PF(t,5) ;
     od:
     a ;
end:
# R. J. Mathar, Jun 24 2009

PF[n_, p_] := p^IntegerExponent[n, p];
usigma[n_] := If[n == 1, 1, Times @@ (1+Power @@@ FactorInteger[n])];
A146891[n_] := Module[{b, a, k, t},
   b = {2^n};
   While[b[[-1]] != 1,
      t = usigma[b[[-1]]];
      b = Append[b, t/PF[t, 2]/PF[t, 3]/PF[t, 5]]];
   a = 2^n;
   For[k = 2, k <= Length[b], k++,
      t = usigma[b[[k-1]]];
      a = a/PF[t, 2]*PF[t, 3]*PF[t, 5]];
   a];
Table[A146891[n], {n, 0, 32}] (* Jean-François Alcover, Apr 09 2024, after R. J. Mathar *)

More terms from R. J. Mathar, Jun 24 2009
Edited by R. J. Mathar, Jul 02 2009
Description of relation between a(n) and c(k) corrected by R. J. Mathar, Jul 07 2009

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions