A244098 Total number of divisors of all the ordered prime factorizations of an integer.

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 9, 2, 5, 5, 5, 2, 9, 2, 9, 5, 5, 2, 14, 3, 5, 4, 9, 2, 16, 2, 6, 5, 5, 5, 19, 2, 5, 5, 14, 2, 16, 2, 9, 9, 5, 2, 20, 3, 9, 5, 9, 2, 14, 5, 14, 5, 5, 2, 35, 2, 5, 9, 7, 5, 16, 2, 9, 5, 16, 2, 34, 2, 5, 9, 9, 5, 16, 2, 20, 5, 5

Offset

1,2

Comments

a(n) = total number of ordered prime factorizations dividing all possible ordered prime factorizations making up n.

Example: for n = 12; a(12) = 9 because 12 = 2*2*3 = 2*3*2 = 3*2*2 the divisors of which are 1, 2, 3, 2*2, 2*3, 3*2, 2*2*3, 2*3*2, 3*2*2. This makes 9 ordered prime factorizations dividing all those making up 12.

Dirichlet convolution of A008480 with A000012.

Links

Pierre-Louis Giscard, Table of n, a(n) for n = 1..5000

Formula

Dirichlet generating function: Zeta(s)/(1-P(s)) with Zeta(s) the Riemann zeta function and P(s) the prime zeta function.

G.f. A(x) satisfies: A(x) = x / (1 - x) + Sum_{k>=1} A(x^prime(k)). - Ilya Gutkovskiy, May 30 2020

Example

For n = 6; a(6) = 5 because 6 = 2*3 = 3*2, the divisors of which are 1, 2, 3, 2*3, 3*2. This makes 5 ordered prime factorizations dividing all those making up 6.
For n = 12; a(12) = 9 because 12 = 2*2*3 = 2*3*2 = 3*2*2, the divisors of which are 1, 2, 3, 2*2, 2*3, 3*2, 2*2*3, 2*3*2, 3*2*2. This makes 9 ordered prime factorizations dividing all those making up 12.
For n prime, a(n) = 2 because a prime n has a single ordered prime factorization n with divisors 1 and n. This makes two ordered prime factorizations dividing that making up n.

Mathematica

f[s_]=Zeta[s]/(1-PrimeZetaP[s]); (* Dirichlet g.f *)
(* or *)
Clear[a, b];
a = Prepend[
   Array[Multinomial @@ Last[Transpose[FactorInteger[#]]] &, 200, 2],
   1];
b = Table[1, {u, 1, Length[a]}];
Table[Sum[If[IntegerQ[p/n], b[[n]] a[[p/n]], 0], {n, 1, p}], {p, 1,
  Length[a]}]

Crossrefs

Cf. A000012, A008480.

Sequence in context: A018892 A343654 A100565 * A285573 A325339 A010846

Adjacent sequences: A244095 A244096 A244097 * A244099 A244100 A244101

Keyword

nonn

Author

Pierre-Louis Giscard, Jun 20 2014

Status

approved