A339219 Number of partitions of n into nonprime parts where every part appears at least 2 times.

1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 4, 2, 4, 4, 6, 4, 8, 6, 11, 8, 11, 11, 17, 11, 19, 18, 25, 20, 32, 26, 42, 32, 46, 43, 63, 47, 72, 66, 90, 74, 110, 94, 137, 115, 155, 145, 203, 161, 235, 212, 283, 244, 339, 298, 413, 356, 472, 437, 589, 496, 681, 625, 810, 718, 962

Offset: 0

Author

Ilya Gutkovskiy, Nov 27 2020

Keywords

• nonn

Examples

			a(12) = 4 because we have [6, 6], [4, 4, 4], [4, 4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A002095, A007690, A018252, A161077, A339218.

Programs

Mathematica

nmax = 66; CoefficientList[Series[Product[1 + Boole[!PrimeQ[k]] x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions[n],?(NoneTrue[#,PrimeQ]&&Min[Length/@Split[#]]>1&)],{n,0,70}] (* _Harvey P. Dale, Oct 03 2024 *)

Formula

G.f.: Product_{k>=1} (1 + x^(2*A018252(k)) / (1 - x^A018252(k))).