A282290 - OEIS

%I #7 Feb 16 2025 08:33:41

%S 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,

%T 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,

%U 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

%N 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).

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

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

%H Ilya Gutkovskiy, <a href="/A282290/a282290.pdf">Extended graphical example</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

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

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

%t 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]

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

%K nonn,changed

%O 0,11

%A _Ilya Gutkovskiy_, Feb 11 2017