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A342621 Sum of the partition number of the prime factors of n with multiplicity. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A072275

Adjacent sequences:  A342618 A342619 A342620 * A342622 A342623 A342624

KEYWORD

nonn

AUTHOR

Eric Desbiaux, Mar 16 2021

STATUS

approved

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Last modified November 28 03:29 EST 2021. Contains 349400 sequences. (Running on oeis4.)