%I #12 Aug 06 2017 21:46:31
%S 1,6,20,72,72,72,20,72,72,17280,4800,17280,72,17280,1152000,5184,5184,
%T 5184,96000,5184,345600,1244160,320000,1244160,82944000,89579520,
%U 71663616000,298598400,1244160,82944000,23040000,82944000,19906560000
%N Terminal point of a repeated reduction of usigma starting at 2^n.
%C 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.
%C Multi-indexed PF_(p1,p2,...)(n) are defined as the products PF_(p1)(n)*PF_(p2)(n)*...
%C 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.
%C The sequence is defined by a(n) = c(k), the auxiliary sequence c at the point where b terminates.
%C All values of the sequence a(n) are 5-smooth, i.e., members of A051037.
%e n=5
%e b(n) : 2^5 -> 11 -> 1
%e c(n) : 2^5 -> 2^5*3 -> 2^3*3^2
%e So a(5) = c(2) = 2^3*3^2 = 72.
%p 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:
%p A146891 := proc(n) local b,a,k,t ;
%p b := [2^n] ;
%p while op(-1,b) <> 1 do
%p t := A034448(op(-1,b)) ;
%p b := [op(b), t/A006519(t)/ A038500(t)/PF(t,5) ] ;
%p od:
%p a := 2^n ;
%p for k from 2 to nops(b) do
%p t := A034448(op(k-1,b)) ;
%p a := a/ A006519(t) *A038500(t)*PF(t,5) ;
%p od:
%p a ;
%p end:
%p # _R. J. Mathar_, Jun 24 2009
%Y Cf. A146892, A151659.
%K nonn
%O 0,2
%A _Yasutoshi Kohmoto_, Apr 17 2009
%E More terms from _R. J. Mathar_, Jun 24 2009
%E Edited by _R. J. Mathar_, Jul 02 2009
%E Description of relation between a(n) and c(k) corrected by _R. J. Mathar_, Jul 07 2009
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