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

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, 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, 132, 1, 5, 18, 112, 5, 31, 1, 18, 5, 31, 1, 264, 1, 5, 18, 18, 5

Offset: 1

Author

Geoffrey Critzer, Feb 01 2015

Keywords

• nonn

Comments

What is the limit log(Sum_{k=1..n} a(k)) / log(n) ?. - Vaclav Kotesovec, Feb 03 2019

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph log(Sum_{k=1..n} a(k)) / log(n), 10^8 terms

Crossrefs

Cf. A074206.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; expand(x*(1+
      add(b(n/d), d=divisors(n) minus {1, n})))
    end:
a:= n-> (p-> add(coeff(p, x, i)*i, i=1..degree(p)))(b(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 01 2015

Mathematica

f[n_] := f[n] =Level[Table[Map[Prepend[#, d] &, f[n/d]], {d,Rest[Divisors[n]]}], {2}];
f[1] = {{}};
g[list_] := Sum[list[[i]] i, {i, 1, Length[list]}];
Prepend[Rest[Map[g,Map[Table[Count[#, i], {i, 1, Max[#]}] &,Map[Length, Map[Sort, Table[f[n], {n, 1, 60}]], {2}]]]], 1]

Formula

Dirichlet generating function: zeta(s)/(1 - zeta(s))^2.

a(n) = Sum_{k>=1} A251683(n,k)*k.