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

0,9

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);
C = [0, 0, C(1:N-1)] % Robert Israel, Feb 12 2017

Crossrefs

Cf. A001221, A000469, A008683, A098983.

Sequence in context: A094718 A076191 A362955 * A286971 A025861 A090723

Adjacent sequences: A282315 A282316 A282317 * A282319 A282320 A282321

Keyword

nonn,look

Author

Ilya Gutkovskiy, Feb 11 2017

Status

approved