A363241 Number of partitions of n with prime rank.

0, 0, 1, 1, 1, 3, 3, 6, 6, 10, 12, 19, 22, 33, 38, 54, 65, 91, 106, 145, 173, 228, 274, 356, 424, 545, 652, 823, 986, 1232, 1468, 1822, 2172, 2665, 3173, 3869, 4590, 5568, 6591, 7938, 9386, 11249, 13256, 15821, 18608, 22100, 25941, 30695, 35933, 42373, 49501, 58160, 67814, 79434, 92396, 107932

Offset: 1

Author

Seiichi Manyama, May 23 2023

Keywords

• nonn
• easy

Examples

			a(6) = 3 counts these partitions: 6, 5+1, 4+2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Wikipedia, Rank of a partition

Crossrefs

Cf. A000041, A010051, A037032, A063995.

Programs

Maple

b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(isprime(i-c), 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..56);  # Alois P. Heinz, May 23 2023

Mathematica

b[n_, i_, c_] := b[n, i, c] = If[i > n, 0, If[i == n, If[i-c > 0 && PrimeQ[i-c], 1, 0], b[n-i, i, c+1] + b[n, i+1, c]]];
a[n_] := b[n, 1, 1];
Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Dec 20 2024, after Alois P. Heinz *)

PARI

my(N=60, x='x+O('x^N)); concat([0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*sum(j=1, N, isprime(j)*x^(k*j)))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * Sum_{p prime} x^(k*p).

a(n) = Sum_{p prime} A063995(n,p). - Alois P. Heinz, Dec 20 2024