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
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
Programs
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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
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