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

%I A146891 #15 Apr 09 2024 08:39:08
%S A146891 1,6,20,72,72,72,20,72,72,17280,4800,17280,72,17280,1152000,5184,5184,
%T A146891 5184,96000,5184,345600,1244160,320000,1244160,82944000,89579520,
%U A146891 71663616000,298598400,1244160,82944000,23040000,82944000,19906560000
%N A146891 Terminal point of a repeated reduction of usigma starting at 2^n.
%C A146891 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 A146891 Multi-indexed PF_(p1,p2,...)(n) are defined as the products PF_(p1)(n)*PF_(p2)(n)*...
%C A146891 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 A146891 The sequence is defined by a(n) = c(k), the auxiliary sequence c at the point where b terminates.
%C A146891 All values of the sequence a(n) are 5-smooth, i.e., members of A051037.
%e A146891 n=5
%e A146891 b(n) : 2^5 -> 11 -> 1
%e A146891 c(n) : 2^5 -> 2^5*3 -> 2^3*3^2
%e A146891 So a(5) = c(2) = 2^3*3^2 = 72.
%p A146891 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 A146891 := proc(n) local b,a,k,t ;
%p A146891     b := [2^n] ;
%p A146891     while op(-1,b) <> 1 do
%p A146891         t := A034448(op(-1,b)) ;
%p A146891         b := [op(b), t/A006519(t)/ A038500(t)/PF(t,5) ] ;
%p A146891     od:
%p A146891     a := 2^n ;
%p A146891     for k from 2 to nops(b) do
%p A146891         t := A034448(op(k-1,b)) ;
%p A146891         a := a/ A006519(t) *A038500(t)*PF(t,5) ;
%p A146891      od:
%p A146891      a ;
%p A146891 end:
%p A146891 # _R. J. Mathar_, Jun 24 2009
%t A146891 PF[n_, p_] := p^IntegerExponent[n, p];
%t A146891 usigma[n_] := If[n == 1, 1, Times @@ (1+Power @@@ FactorInteger[n])];
%t A146891 A146891[n_] := Module[{b, a, k, t},
%t A146891    b = {2^n};
%t A146891    While[b[[-1]] != 1,
%t A146891       t = usigma[b[[-1]]];
%t A146891       b = Append[b, t/PF[t, 2]/PF[t, 3]/PF[t, 5]]];
%t A146891    a = 2^n;
%t A146891    For[k = 2, k <= Length[b], k++,
%t A146891       t = usigma[b[[k-1]]];
%t A146891       a = a/PF[t, 2]*PF[t, 3]*PF[t, 5]];
%t A146891    a];
%t A146891 Table[A146891[n], {n, 0, 32}] (* _Jean-François Alcover_, Apr 09 2024, after _R. J. Mathar_ *)
%Y A146891 Cf. A146892, A151659.
%K A146891 nonn
%O A146891 0,2
%A A146891 _Yasutoshi Kohmoto_, Apr 17 2009
%E A146891 More terms from _R. J. Mathar_, Jun 24 2009
%E A146891 Edited by _R. J. Mathar_, Jul 02 2009
%E A146891 Description of relation between a(n) and c(k) corrected by _R. J. Mathar_, Jul 07 2009