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A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

Conjecture: a(n) > 0 for all n > 108.

LINKS

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

FORMULA

G.f.: (Sum_{k>=1} x^A246547(k))*(Sum_{k>=1} x^A000469(k)).

EXAMPLE

a(26) = 3 because we have [25, 1], [22, 4] and [16, 10].

MATHEMATICA

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]

PROG

(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

CROSSREFS

Cf. A000469, A098983, A246547, A282192, A282290, A282318, A282947.

Sequence in context: A024941 A219492 A285796 * A100699 A108921 A071548

Adjacent sequences:  A287296 A287297 A287298 * A287300 A287301 A287302

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 23 2017

STATUS

approved

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Last modified February 22 06:17 EST 2018. Contains 299430 sequences. (Running on oeis4.)