

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).


3



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
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OFFSET

0,11


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.


LINKS

Table of n, a(n) for n=0..110.
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree


FORMULA

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


EXAMPLE

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


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]


CROSSREFS

Cf. A005117, A008683, A098983, A144338, A246547.
Sequence in context: A270823 A067627 A077233 * A178795 A123185 A133569
Adjacent sequences: A282287 A282288 A282289 * A282291 A282292 A282293


KEYWORD

nonn


AUTHOR

Ilya Gutkovskiy, Feb 11 2017


STATUS

approved



