A339241 Number of partitions of n into prime power parts (including 1) where every part appears at least 2 times.
1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 15, 11, 21, 19, 31, 27, 46, 40, 63, 60, 88, 83, 124, 117, 166, 165, 224, 222, 303, 301, 399, 407, 525, 537, 691, 707, 893, 929, 1153, 1202, 1485, 1550, 1890, 1992, 2400, 2534, 3040, 3212, 3818, 4059, 4781, 5089, 5972, 6359, 7412
Offset: 0
Keywords
Examples
a(6) = 4 because we have [3, 3], [2, 2, 2], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].
Programs
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Mathematica
nmax = 56; CoefficientList[Series[(1 + x^2/(1 - x)) Product[1 + Boole[PrimePowerQ[k]] x^(2 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Formula
G.f.: (1 + x^2 / (1 - x)) * Product_{p prime, k>=1} (1 + x^(2*p^k) / (1 - x^(p^k))).