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

1,6

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Wikipedia, Rank of a partition

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

Example

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

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 *)

Prog

(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)))))

Crossrefs

Cf. A000041, A010051, A037032, A063995.

Sequence in context: A237665 A355394 A088528 * A220153 A219627 A340571

Adjacent sequences: A363238 A363239 A363240 * A363242 A363243 A363244

Keyword

nonn,easy,changed

Author

Seiichi Manyama, May 23 2023

Status

approved