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A339221
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Number of partitions of n into cubes where every part appears at least 2 times.
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0
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1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 11, 10, 11, 11, 11, 11, 12, 11, 13, 13, 13, 14, 14, 14, 15, 14, 16, 15, 16, 16, 16, 16, 17, 16
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OFFSET
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0,17
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LINKS
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FORMULA
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G.f.: Product_{k>=1} (1 + x^(2*k^3) / (1 - x^(k^3))).
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EXAMPLE
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a(24) = 3 because we have [8, 8, 8], [8, 8, 1, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
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MATHEMATICA
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nmax = 95; CoefficientList[Series[Product[1 + x^(2 k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n], AllTrue[Surd[#, 3], IntegerQ] && Min[ Tally[ #][[All, 2]]]>1&]], {n, 0, 100}] (* Requires Mathematica version 10 or later *) (* The program takes a long time to run. *) (* Harvey P. Dale, May 03 2021 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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