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

0,31

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

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).

Example

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

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 *)

Crossrefs

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

Sequence in context: A159818 A081827 A100286 * A332662 A364036 A358580

Adjacent sequences: A280909 A280910 A280911 * A280913 A280914 A280915

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 10 2017

Status

approved