

A282318


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


4



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
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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:N1)] % Robert Israel, Feb 12 2017


CROSSREFS

Cf. A001221, A000469, A008683, A098983.
Sequence in context: A053616 A094718 A076191 * A286971 A025861 A090723
Adjacent sequences: A282315 A282316 A282317 * A282319 A282320 A282321


KEYWORD

nonn,look


AUTHOR

Ilya Gutkovskiy, Feb 11 2017


STATUS

approved



