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A280912
Number of partitions of n into odd semiprimes (A046315).
1
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
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
KEYWORD
nonn,changed
AUTHOR
Ilya Gutkovskiy, Jan 10 2017
STATUS
approved