A394668 Number of partitions of n^2 into prime parts.

1, 0, 1, 4, 14, 52, 197, 744, 2833, 10776, 40899, 154709, 582925, 2186867, 8166824, 30356317, 112302573, 413501182, 1515404880, 5528096761, 20075018700, 72579401227, 261273265048, 936590133846, 3343695225246, 11889828821851, 42115784612366, 148621625010955, 522556189965294

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

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

Example

For n = 2, a(2) = 1 because n^2 = 4 can be partitioned into primes in exactly 1 way:
  4 = 2 + 2.
For n = 3, a(3) = 4 because n^2 = 9 has exactly 4 partitions into prime parts:
  9 = 7 + 2, 9 = 5 + 2 + 2, 9 = 3 + 3 + 3, 9 = 3 + 2 + 2 + 2.

Mathematica

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

Prog

(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

Crossrefs

Subsequence of A000607.

Cf. A000290.

Sequence in context: A047033 A382320 A017947 * A244898 A244935 A199698

Adjacent sequences: A394664 A394666 A394667 * A394669 A394670 A394671

Keyword

nonn

Author

Michael Shmoish, May 02 2026

Extensions

More terms from Jason Yuen, May 03 2026

Status

approved