A024941 Number of partitions of n into distinct primes of the form 4k + 1.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 3, 0, 1, 0, 1, 1, 0, 3, 1, 0, 0, 2, 3, 1, 1, 1, 2, 1, 0, 3, 2, 0, 0, 1, 5, 1, 0, 1, 2, 3, 1, 3, 3, 1, 0, 2, 5, 3, 1, 1, 2, 3, 2, 4, 4, 1, 1, 2, 7, 4, 1, 2, 3

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

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

Examples

			a(41) = 1 since it can be expressed as a sum of primes of the form 4k + 1 in only one way: a trivial partition containing just itself.
a(42) = 2 since 42 = 5 + 37 = 13 + 29.
Although 43 = 2 * 13 + 17 = 6 * 5 + 13, none of those consist of distinct primes only. Hence a(43) = 0.
		

Links

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

Crossrefs

Cf. A024942 (4k - 1).

Programs

Mathematica

searchMax = 120; primes4kp1 = Select[4Range[Floor[searchMax/4]] + 1, PrimeQ]; Table[Length[Select[IntegerPartitions[n, All, primes4kp1], DuplicateFreeQ]], {n, 0, searchMax}] (* Alonso del Arte, Apr 17 2019 *)

PARI

{ my(V=select(x->x%4==1,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