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A244098 Total number of divisors of all the ordered prime factorizations of an integer. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A330833 A018892 A100565 * A285573 A325339 A010846

Adjacent sequences:  A244095 A244096 A244097 * A244099 A244100 A244101

KEYWORD

nonn

AUTHOR

Pierre-Louis Giscard, Jun 20 2014

STATUS

approved

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Last modified August 13 00:27 EDT 2020. Contains 336441 sequences. (Running on oeis4.)