A340756 Number of partitions of n into 4 semiprime parts.

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 3, 3, 4, 2, 3, 4, 5, 6, 5, 4, 7, 7, 9, 9, 9, 7, 9, 12, 13, 11, 11, 13, 16, 17, 17, 18, 18, 17, 20, 25, 25, 23, 24, 26, 32, 29, 31, 33, 31, 33, 35, 43, 43, 40, 39, 45, 48, 52, 50, 52, 53, 52, 61, 69, 67, 61, 61, 70, 79, 76, 76, 80, 81, 85, 88, 101

Offset: 16

Author

Wesley Ivan Hurt, Jan 19 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 16..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A001222 (Omega), A001358.

Column k=4 of A344447.

Programs

Mathematica

Table[Sum[Sum[Sum[KroneckerDelta[PrimeOmega[k], PrimeOmega[j], PrimeOmega[i], PrimeOmega[n - i - j - k], 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 16, 100}]
Table[Count[IntegerPartitions[n,{4}],?(PrimeOmega[#]=={2,2,2,2}&)],{n,16,95}] (* _Harvey P. Dale, May 14 2022 *)

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} [Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^4] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021