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A254577 Total number of factors over all ordered factorizations of n.

%I A254577 #13 Feb 03 2019 09:03:41
%S A254577 1,1,1,3,1,5,1,8,3,5,1,18,1,5,5,20,1,18,1,18,5,5,1,56,3,5,8,18,1,31,1,
%T A254577 48,5,5,5,75,1,5,5,56,1,31,1,18,18,5,1,160,3,18,5,18,1,56,5,56,5,5,1,
%U A254577 132,1,5,18,112,5,31,1,18,5,31,1,264,1,5,18,18,5
%N A254577 Total number of factors over all ordered factorizations of n.
%C A254577 What is the limit log(Sum_{k=1..n} a(k)) / log(n) ?. - _Vaclav Kotesovec_, Feb 03 2019
%H A254577 Alois P. Heinz, <a href="/A254577/b254577.txt">Table of n, a(n) for n = 1..10000</a>
%H A254577 Vaclav Kotesovec, <a href="/A254577/a254577.jpg">Graph log(Sum_{k=1..n} a(k)) / log(n), 10^8 terms</a>
%F A254577 Dirichlet generating function: zeta(s)/(1 - zeta(s))^2.
%F A254577 a(n) = Sum_{k>=1} A251683(n,k)*k.
%e A254577 a(20)=18 because in the ordered factorizations of twenty: 20, 2*10, 4*5, 5*4, 10*2, 2*2*5, 2*5*2, 5*2*2 there are a total of 18 factors.
%p A254577 with(numtheory):
%p A254577 b:= proc(n) option remember; expand(x*(1+
%p A254577       add(b(n/d), d=divisors(n) minus {1, n})))
%p A254577     end:
%p A254577 a:= n-> (p-> add(coeff(p, x, i)*i, i=1..degree(p)))(b(n)):
%p A254577 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 01 2015
%t A254577 f[n_] := f[n] =Level[Table[Map[Prepend[#, d] &, f[n/d]], {d,Rest[Divisors[n]]}], {2}];
%t A254577 f[1] = {{}};
%t A254577 g[list_] := Sum[list[[i]] i, {i, 1, Length[list]}];
%t A254577 Prepend[Rest[Map[g,Map[Table[Count[#, i], {i, 1, Max[#]}] &,Map[Length, Map[Sort, Table[f[n], {n, 1, 60}]], {2}]]]], 1]
%Y A254577 Cf. A074206.
%K A254577 nonn
%O A254577 1,4
%A A254577 _Geoffrey Critzer_, Feb 01 2015