A316202 Number of integer partitions of n into Fermi-Dirac primes.

1, 0, 1, 1, 2, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 31, 37, 45, 54, 65, 77, 92, 109, 128, 152, 177, 208, 242, 283, 327, 380, 439, 506, 583, 669, 768, 878, 1004, 1144, 1303, 1482, 1681, 1906, 2156, 2438, 2750, 3101, 3490, 3924, 4407, 4942, 5538, 6197, 6929

Offset: 0

Author

Gus Wiseman, Jun 26 2018

Keywords

• nonn

Comments

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0.

Examples

			The a(12) = 13 integer partitions of 12 into Fermi-Dirac primes:
(75), (93),
(444), (543), (552), (732),
(3333), (4332), (4422), (5322),
(33222), (42222),
(222222).
		

Crossrefs

Cf. A000586, A000607, A050376, A064547, A213925, A279065, A299755, A299757, A305829.

Programs

Mathematica

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]]
FDprimeList=Select[Range[nn],FDpQ];
ser=Product[1/(1-x^d),{d,FDprimeList}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

Formula

O.g.f.: Product_d 1/(1 - x^d) where the product is over all Fermi-Dirac primes (A050376).