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

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

Author

Yasutoshi Kohmoto, Apr 17 2009

Keywords

• nonn

Comments

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.

Examples

			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.
		

Crossrefs

Cf. A146892, A151659.

Programs

Maple

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

Mathematica

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 *)

Extensions

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