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A287299
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Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).
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1
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0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 3, 0, 1, 1, 4, 3, 0, 2, 2, 2, 0, 3, 4, 3, 1, 2, 6, 3, 1, 0, 5, 4, 2, 2, 4, 3, 0, 2, 3, 5, 0, 1, 3, 4, 3, 2, 4, 3, 3, 4, 5, 4, 0, 2, 5, 5, 0, 4, 6, 2, 1, 1, 7, 3, 1, 2, 7, 4, 2, 4, 5, 5, 1, 3, 6, 5, 1, 3, 6, 6, 3, 4, 4, 4, 2, 4, 7, 6, 3, 1, 4, 4, 0, 4, 6, 5, 2, 2, 7, 5, 2, 1, 7, 8, 4
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OFFSET
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0,11
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COMMENTS
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Conjecture: a(n) > 0 for all n > 108.
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LINKS
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Table of n, a(n) for n=0..120.
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree
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FORMULA
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G.f.: (Sum_{k>=1} x^A246547(k))*(Sum_{k>=1} x^A000469(k)).
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EXAMPLE
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a(26) = 3 because we have [25, 1], [22, 4] and [16, 10].
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MATHEMATICA
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nmax = 120; CoefficientList[Series[(Sum[Boole[SquareFreeQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]) (Sum[Boole[PrimePowerQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
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PROG
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(PARI) x='x+O('x^120); concat([0, 0, 0, 0, 0], Vec(sum(k=1, 120, (issquarefree(k) && !isprime(k))*x^k) * sum(k=1, 120, (isprimepower(k) && !isprime(k))*x^k))) \\ Indranil Ghosh, May 23 2017
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CROSSREFS
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Cf. A000469, A098983, A246547, A282192, A282290, A282318, A282947.
Sequence in context: A219492 A285796 A317241 * A100699 A108921 A071548
Adjacent sequences: A287296 A287297 A287298 * A287300 A287301 A287302
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, May 23 2017
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STATUS
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approved
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