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

1,4

Comments

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

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

Formula

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

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

Example

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.

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]

Crossrefs

Cf. A074206.

Sequence in context: A029669 A050329 A147005 * A051707 A302787 A240535

Adjacent sequences: A254574 A254575 A254576 * A254578 A254579 A254580

Keyword

nonn

Author

Geoffrey Critzer, Feb 01 2015

Status

approved