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

1,2

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000041, A027746, A058698, A001222.

Sequence in context: A245703 A260426 A167151 * A273014 A336321 A359446

Adjacent sequences: A342618 A342619 A342620 * A342622 A342623 A342624

Keyword

nonn

Author

Eric Desbiaux, Mar 16 2021

Status

approved