A282318 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

Ilya Gutkovskiy, Extended graphical example

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).

EXAMPLE

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

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]

PROG

(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);