A280912 Number of partitions of n into odd semiprimes (A046315).

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 2, 1, 1, 2, 0, 0, 3, 1, 0, 3, 1, 1, 3, 1, 0, 4, 2, 2, 5, 1, 1, 5, 3, 1, 6, 3, 2, 8, 2, 1, 7, 5, 4, 9, 4, 3, 11, 6, 3, 11, 6, 6, 14, 7, 5, 15, 9, 7, 16, 9, 8, 20, 14, 9, 21, 13, 11, 26, 16, 12, 28, 19, 17, 29, 19, 17, 37, 27

Offset: 0

Author

Ilya Gutkovskiy, Jan 10 2017

Keywords

• nonn

Examples

			a(39) = 3 because we have [39], [21, 9, 9] and [15, 15, 9].
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Semiprime
• Index entries for related partition-counting sequences

Crossrefs

Cf. A001222, A001358, A046315, A099773, A101048, A112020.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1 - Floor[PrimeOmega[2 k + 1]/2] Floor[2/PrimeOmega[2 k + 1]] x^(2 k + 1)), {k, 1, nmax}], {x, 0, nmax}], x]
Join[{1},Table[Count[IntegerPartitions[n],?(AllTrue[#,OddQ]&&Union[PrimeOmega[#]]=={2}&)],{n,110}]] (* _Harvey P. Dale, Nov 11 2024 *)

Formula

G.f.: Product_{k>=1} 1/(1 - floor(bigomega(2*k+1)/2)*floor(2/bigomega(2*k+ 1))*x^(2*k+1)), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).