A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

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

Offset: 0

Author

Ilya Gutkovskiy, May 23 2017

Keywords

• nonn

Comments

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

Examples

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

Links

• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Prime Power
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

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

Programs

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]

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

Formula

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