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%I #17 Dec 17 2017 03:15:20
%S 0,0,0,1,1,0,1,0,2,1,0,1,2,2,1,1,1,4,2,2,2,2,1,3,3,3,2,3,3,4,1,2,4,5,
%T 2,4,2,6,5,4,4,6,3,5,6,6,4,5,3,6,3,6,5,8,3,4,4,7,6,6,4,5,8,6,6,7,2,7,
%U 9,8,5,7,6,8,8,8,8,9,3,8,9,10,8,8,5,10,6,9,10,13,4,6,8,12,10,9,8,10,12,10,9,9,7,8,11,12,9,10
%N Number of ways of writing n as a sum of a prime and a nonprime squarefree number.
%C Conjecture: a(n) > 0 for all n > 10.
%H Robert Israel, <a href="/A282318/b282318.txt">Table of n, a(n) for n = 0..10000</a>
%H Ilya Gutkovskiy, <a href="/A282318/a282318.pdf">Extended graphical example</a>
%F G.f.: (Sum_{i>=1} x^prime(i))*(x + Sum_{j>=2} sgn(omega(j)-1)*mu(j)^2*x^j), where omega(j) is the number of distinct primes dividing j (A001221) and mu(j) is the Moebius function (A008683).
%e a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].
%t nmax = 107; CoefficientList[Series[(Sum[x^Prime[i], {i, 1, nmax}]) (x + Sum[Sign[PrimeNu[j] - 1] MoebiusMu[j]^2 x^j, {j, 2, nmax}]), {x, 0, nmax}], x]
%o (MATLAB)
%o N = 200; % to get a(0) to a(N)
%o Primes = primes(N);
%o B = zeros(1,N);
%o B(Primes) = 1;
%o LPrimes = Primes(Primes .^ 2 < N);
%o SF = 1 - B;
%o for p = LPrimes
%o SF(p^2:p^2:N) = 0;
%o end
%o C = conv(SF, B);
%o C = [0,0,C(1:N-1)] % _Robert Israel_, Feb 12 2017
%Y Cf. A001221, A000469, A008683, A098983.
%K nonn,look
%O 0,9
%A _Ilya Gutkovskiy_, Feb 11 2017
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