A342621 Sum of the partition number of the prime factors of n with multiplicity.

0, 2, 3, 4, 7, 5, 15, 6, 6, 9, 56, 7, 101, 17, 10, 8, 297, 8, 490, 11, 18, 58, 1255, 9, 14, 103, 9, 19, 4565, 12, 6842, 10, 59, 299, 22, 10, 21637, 492, 104, 13, 44583, 20, 63261, 60, 13, 1257, 124754, 11, 30, 16, 300, 105, 329931, 11, 63, 21, 493, 4567, 831820

Offset: 1

Author

Eric Desbiaux, Mar 16 2021

Keywords

• nonn

Examples

			For n = 408 = 2^3*3*17, a(408) = 3 * A000041(2) + A000041(3) + A000041(17) = 3*2 + 3 + 297 = 306.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 3000 terms from Eric Desbiaux)

Crossrefs

Cf. A000041, A027746, A058698, A001222.

Programs

Maple

a:= n-> add(combinat[numbpart](i[1])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 17 2021

Mathematica

{0}~Join~Array[Total@ Map[#2 PartitionsP[#1] & @@ # &, FactorInteger[#]] &, 58, 2] (* Michael De Vlieger, Mar 17 2021 *)

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]*numbpart(f[k,1])); \\ Michel Marcus, Mar 17 2021

Sage

def a(n):
    return sum([Partitions(primefactor).cardinality() for (primefactor,exponent) in factor(n) for _ in range(exponent)])
[a(n) for n in (1..100)]

Formula

a(A003586(n)) - A001414(A003586(n)) = 0.

a(A006899(n)) * A008480(A006899(n)) - A001414(A006899(n)) = 0.

a(n) = Sum_{k=1..A001222(n)} A000041(A027746(n,k)). - Alois P. Heinz, Apr 09 2021