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

Author

Gus Wiseman, Feb 05 2018

Keywords

• nonn
• look
• mult

Comments

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

Examples

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

Links

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

Crossrefs

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

Cf. A000720, A003964.

Programs

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}]

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

Formula

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