A282290 Expansion of (Sum_{p prime, i>=2} x^(p^i))*(Sum_{j>=2} mu(j)^2*x^j), where mu() is the Moebius function (A008683).

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 4, 1, 3, 3, 4, 1, 2, 3, 4, 2, 3, 6, 4, 3, 3, 4, 5, 1, 5, 7, 6, 3, 3, 7, 4, 3, 4, 7, 6, 3, 4, 5, 7, 2, 3, 5, 7, 4, 3, 4, 5, 4, 4, 7, 6, 4, 4, 8, 6, 4, 6, 7, 7, 2, 5, 7, 7, 2, 4, 9, 5, 4, 4, 7, 8, 4, 5, 9, 9, 4, 4, 7, 7, 5, 6, 8, 8, 5, 5, 8, 6, 4, 6, 8, 7, 5, 6, 6, 6, 2, 5, 10

Offset: 0

Author

Ilya Gutkovskiy, Feb 11 2017

Keywords

• nonn

Comments

Number of ways of writing n as a sum of a proper prime power (A246547) and a squarefree number > 1 (A144338).

Conjecture: a(n) > 0 for all n > 8.

Examples

			a(19) = 4 because we have [16, 3], [15, 4], [11, 8] and [10, 9].
		

Links

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

Crossrefs

Cf. A005117, A008683, A098983, A144338, A246547.

Programs

Mathematica

nmax = 110; CoefficientList[Series[Sum[Sign[PrimeOmega[i] - 1] Floor[1/PrimeNu[i]] x^i, {i, 2, nmax}] Sum[MoebiusMu[j]^2 x^j, {j, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (Sum_{p prime, i>=2} x^(p^i))*(Sum_{j>=2} mu(j)^2*x^j).