A382871 Number of ways to partition distinct prime numbers into two disjoint sets such that the sum of each set equals n.

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 3, 3, 2, 3, 4, 6, 2, 5, 0, 5, 9, 7, 14, 8, 6, 10, 9, 21, 19, 11, 18, 15, 29, 34, 35, 34, 24, 31, 51, 55, 48, 76, 34, 60, 93, 89, 97, 91, 76, 83, 156, 164, 189, 145, 157, 172, 186, 283, 276, 218, 242, 280, 405, 433, 476, 446

Offset: 0

Author

Seiichi Manyama, Apr 09 2025

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for n > 27.

Examples

			a(18) = 2: [13, 5; 11, 7], [13, 3, 2; 11, 7].
a(19) = 3: [19; 17, 2], [19; 11, 5, 3], [17, 2; 11, 5, 3].
		

Crossrefs

Cf. A000607, A108796, A382954.

Programs

PARI

a(n) = my(x='x+O('x^(n+1)), y='y+O('y^(n+1)));if(n==0, 1, polcoef(polcoef(prod(k=1, n, 1+x^prime(k)+y^prime(k)), n), n)/2);

Formula

a(n) = 1/2 * [(x*y)^n] Product_{p prime} (1 + x^p + y^p) for n > 0.