A363241 - OEIS

%I #23 Dec 20 2024 11:27:25

%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.

%H Alois P. Heinz, <a href="/A363241/b363241.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rank_of_a_partition">Rank of a partition</a>

%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).

%F a(n) = Sum_{p prime} A063995(n,p). - _Alois P. Heinz_, Dec 20 2024

%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), 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

%t 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]]];

%t a[n_] := b[n, 1, 1];

%t Table[a[n], {n, 1, 56}] (* _Jean-François Alcover_, Dec 20 2024, after _Alois P. Heinz_ *)

%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, A063995.

%K nonn,easy,changed

%O 1,6

%A _Seiichi Manyama_, May 23 2023