A299200 Number of twice-partitions whose domain is the integer partition with Heinz number n.

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 15, 4, 22, 3, 10, 7, 30, 2, 9, 11, 8, 5, 42, 6, 56, 1, 14, 15, 15, 4, 77, 22, 22, 3, 101, 10, 135, 7, 12, 30, 176, 2, 25, 9, 30, 11, 231, 8, 21, 5, 44, 42, 297, 6, 385, 56, 20, 1, 33, 14, 490, 15, 60, 15, 627, 4

Offset

1,3

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(prime(n)) = A000041(n).

Example

The a(15) = 6 twice-partitions: (3)(2), (3)(11), (21)(2), (21)(11), (111)(2), (111)(11).

Maple

with(numtheory): with(combinat):
a:= n-> mul(numbpart(pi(i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..82);  # Alois P. Heinz, Jan 14 2021

Mathematica

Table[Times@@Cases[FactorInteger[n], {p_, k_}:>PartitionsP[PrimePi[p]]^k], {n, 100}]

Prog

(PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = numbpart(primepi(f[k, 1])); ); factorback(f); } \\ Michel Marcus, Feb 26 2018

Crossrefs

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299201, A299202, A299203.

Cf. A000720, A003964.

Sequence in context: A064989 A290099 A250479 * A332819 A321272 A321270

Adjacent sequences: A299197 A299198 A299199 * A299201 A299202 A299203

Keyword

nonn,look,mult

Author

Gus Wiseman, Feb 05 2018

Status

approved