A280683 Number of ways to write n as an ordered sum of two positive squarefree semiprimes (A006881).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 2, 3, 2, 1, 2, 4, 0, 0, 2, 6, 2, 0, 2, 4, 4, 1, 4, 5, 4, 0, 4, 8, 6, 2, 0, 5, 4, 4, 4, 6, 4, 0, 4, 8, 10, 0, 2, 4, 6, 3, 6, 9, 4, 3, 6, 14, 8, 2, 4, 5, 8, 3, 10, 8, 4, 0, 8, 12, 4, 4, 4, 8, 6, 8, 12, 11, 6, 2, 10, 12, 12, 4, 8, 12, 12, 5, 12, 10, 4, 6

Offset: 1

Author

Ilya Gutkovskiy, Jan 07 2017

Keywords

• nonn
• easy

Comments

Conjecture: a(n) > 0 for n > 82 (see comment in A006881 from Richard R. Forberg).

Examples

			a(20) = 3 because we have [14, 6], [10, 10] and [6, 14].
		

Links

• Ilya Gutkovskiy, Extended graphical example
• Eric Weisstein's World of Mathematics, Semiprime
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Cf. A001222, A001358, A005117, A006881, A008683, A073610, A098235, A199331, A280710.

Programs

Mathematica

nmax = 106; Rest[CoefficientList[Series[(Sum[MoebiusMu[k]^2 Floor[PrimeOmega[k]/2] Floor[2/PrimeOmega[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]]

Formula

G.f.: (Sum_{k>=2} mu(k)^2*floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k)^2, where mu(k) is the Moebius function (A008683) and bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A280710. - Wesley Ivan Hurt, Jan 07 2024