login
A254577
Total number of factors over all ordered factorizations of n.
4
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
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
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Feb 01 2015
STATUS
approved