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A282947 Number of ways of writing n as a sum of a perfect power and a squarefree semiprime.

%I A282947 #13 Feb 16 2025 08:33:42
%S A282947 0,0,0,0,0,0,0,1,0,0,1,1,0,0,2,2,1,0,2,2,0,0,3,3,1,1,2,1,0,1,4,3,0,1,
%T A282947 2,3,1,3,3,3,2,2,7,3,1,0,4,5,2,2,3,3,1,2,3,4,1,1,4,5,3,2,4,4,3,3,6,3,
%U A282947 0,2,6,6,0,4,4,3,1,1,7,1,1,2,5,5,2,4,4,6,2,3,6,4,2,3,6,6,4,3,4,4,2,5,6,5,3,1,3,5,0,3,6,3,3,2,6,5,3,1,5,7,5
%N A282947 Number of ways of writing n as a sum of a perfect power and a squarefree semiprime.
%C A282947 Conjecture: a(n) > 0 for all n > 108.
%C A282947 From _Robert G. Wilson v_, Feb 25 2017: (Start)
%C A282947 Conjecture: a(n) > 1 for all n > 604,
%C A282947 Conjecture: a(n) > 2 for all n > 1008, etc.
%C A282947 First occurrence of k: 0, 7, 14, 22, 30, 47, 66, 42, 127, 138, 150, 222, 251, 303, 210, 430, 330, 462, 670, 770, 983, 878, 1038, 1142, 1355, 1482, ... (End)
%H A282947 Robert G. Wilson v, <a href="/A282947/b282947.txt">Table of n, a(n) for n = 0..1000</a>
%H A282947 Ilya Gutkovskiy, <a href="/A282947/a282947.pdf">Extended graphical example</a>
%H A282947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Powers</a>
%H A282947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime</a>
%H A282947 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%F A282947 G.f.: (Sum_{k>=1} x^A001597(k))*(Sum_{k>=1} x^A006881(k)).
%e A282947 a(22) = 3 because we have [21, 1], [16, 6] and [14, 8].
%t A282947 nmax = 120; CoefficientList[Series[(x + Sum[Boole[GCD @@ FactorInteger[k][[All, 2]] > 1] x^k, {k, 2, nmax}]) (Sum[MoebiusMu[k]^2 Floor[2/PrimeOmega[k]] Floor[PrimeOmega[k]/2] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
%Y A282947 Cf. A001597, A006881, A196228, A282192, A282290.
%K A282947 nonn
%O A282947 0,15
%A A282947 _Ilya Gutkovskiy_, Feb 25 2017