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

1,16

Comments

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

Links

Table of n, a(n) for n=1..106.

Ilya Gutkovskiy, Extended graphical example

Eric Weisstein's World of Mathematics, Semiprime

Eric Weisstein's World of Mathematics, Squarefree

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

Example

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

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]]

Crossrefs

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

Sequence in context: A123391 A212172 A275812 * A171871 A076260 A245527

Adjacent sequences: A280680 A280681 A280682 * A280684 A280685 A280686

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jan 07 2017

Status

approved