A282318 Number of ways of writing n as a sum of a prime and a nonprime squarefree number.

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, 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, 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

Offset: 0

Author

Ilya Gutkovskiy, Feb 11 2017

Keywords

• nonn
• look

Comments

Conjecture: a(n) > 0 for all n > 10.

Examples

			a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Ilya Gutkovskiy, Extended graphical example

Crossrefs

Cf. A001221, A000469, A008683, A098983.

Programs

MATLAB

N = 200; % to get a(0) to a(N)
Primes = primes(N);
B = zeros(1,N);
B(Primes) = 1;
LPrimes = Primes(Primes .^ 2 < N);
SF = 1 - B;
for p = LPrimes
   SF(p^2:p^2:N) = 0;
end
C = conv(SF, B);
C = [0,0,C(1:N-1)] % Robert Israel, Feb 12 2017

Mathematica

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]

Formula

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).