A394668 - OEIS

%I #31 May 07 2026 15:44:32

%S 1,0,1,4,14,52,197,744,2833,10776,40899,154709,582925,2186867,8166824,

%T 30356317,112302573,413501182,1515404880,5528096761,20075018700,

%U 72579401227,261273265048,936590133846,3343695225246,11889828821851,42115784612366,148621625010955,522556189965294

%N Number of partitions of n^2 into prime parts.

%H Alois P. Heinz, <a href="/A394668/b394668.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = A000607(n^2).

%e For n = 2, a(2) = 1 because n^2 = 4 can be partitioned into primes in exactly 1 way:

%e   4 = 2 + 2.

%e For n = 3, a(3) = 4 because n^2 = 9 has exactly 4 partitions into prime parts:

%e   9 = 7 + 2, 9 = 5 + 2 + 2, 9 = 3 + 3 + 3, 9 = 3 + 2 + 2 + 2.

%t a[n_]:=Length[IntegerPartitions[n^2,All,Prime[Range[PrimePi[n^2]]]]];Array[a,14,0] (* _James C. McMahon_, May 07 2026 *)

%o (PARI) my(N=1000, x='x+O('x^N), p=1/prod(k=1,N,1-x^prime(k))); vector(sqrtint(N), i, polcoeff(p,(i-1)^2)) \\ _Michel Marcus_, May 07 2026

%Y Subsequence of A000607.

%Y Cf. A000290.

%K nonn

%O 0,4

%A _Michael Shmoish_, May 02 2026

%E More terms from _Jason Yuen_, May 03 2026