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%I #14 May 23 2023 10:46:36
%S 0,0,1,1,1,3,3,6,6,10,12,19,22,33,38,54,65,91,106,145,173,228,274,356,
%T 424,545,652,823,986,1232,1468,1822,2172,2665,3173,3869,4590,5568,
%U 6591,7938,9386,11249,13256,15821,18608,22100,25941,30695,35933,42373,49501,58160,67814,79434,92396,107932
%N Number of partitions of n with prime rank.
%F 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).
%e a(6) = 3 counts these partitions: 6, 5+1, 4+2.
%p b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
%p `if`(isprime(i-c, 7), 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
%p end:
%p a:= n-> b(n, 1$2):
%p seq(a(n), n=1..56); # _Alois P. Heinz_, May 23 2023
%o (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)))))
%Y Cf. A000041, A010051, A037032.
%K nonn,easy
%O 1,6
%A _Seiichi Manyama_, May 23 2023
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