login
A280683
Number of ways to write n as an ordered sum of two positive squarefree semiprimes (A006881).
2
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
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]]
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Jan 07 2017
STATUS
approved