A024942 Number of partitions of n into distinct primes of the form 4k + 3.

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 0, 1, 2, 1, 0, 2, 2, 0, 0, 2, 1, 0, 1, 2, 2, 1, 1, 2, 1, 1, 0, 2, 3, 0, 2, 3, 3, 0, 1, 4, 1, 1, 2, 4, 2, 1, 3, 3, 3, 1, 3, 4, 3, 2, 2, 6, 3, 1, 4, 6, 4, 1, 6, 6, 3, 3, 5, 7, 3, 3, 6, 7, 6, 2, 7, 9, 3, 2, 7, 10, 2, 4, 11, 10, 4, 5, 12, 7

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

a(0) = 1 corresponds to the empty partition {}.

Examples

			a(26) = 2 since 26 = 3 + 23 = 7 + 19.
Even though 27 = (3 * 3) + 7 + 11 = (2 * 3) + (3 * 7) = (9 * 3), there is no partition of 27 into primes of the form 4k - 1 with all parts distinct. Hence a(27) = 0.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A024941 (4k + 1).

Programs

Mathematica

searchMax = 108; primes4km1 = Select[4Range[Ceiling[searchMax/4]] - 1, PrimeQ]; Table[Length[Select[IntegerPartitions[n, All, primes4km1], DuplicateFreeQ]], {n, 0, searchMax}] (* Alonso del Arte, Apr 16 2019 *)

PARI

{ my(V=select(x->x%4==3,primes(40))); my(x='x+O('x^V[#V])); Vec(prod(k=1,#V,1+x^V[k])) } \\ Joerg Arndt, Apr 19 2019

Extensions

Definition clarified by Felix Fröhlich, Apr 17 2019

a(0) = 1 prepended by Joerg Arndt, Apr 19 2019