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

16,7

Links

Alois P. Heinz, Table of n, a(n) for n = 16..10000

Index entries for sequences related to partitions

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

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 *)

Crossrefs

Cf. A001222 (Omega), A001358.

Column k=4 of A344447.

Sequence in context: A029334 A275078 A286478 * A030273 A029197 A029174

Adjacent sequences: A340753 A340754 A340755 * A340757 A340758 A340759

Keyword

nonn

Author

Wesley Ivan Hurt, Jan 19 2021

Status

approved