A280829 Number of partitions of n into two squarefree semiprimes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 1, 2, 0, 0, 1, 3, 1, 0, 1, 2, 2, 1, 2, 3, 2, 0, 2, 4, 3, 1, 0, 3, 2, 2, 2, 3, 2, 0, 2, 4, 5, 0, 1, 2, 3, 2, 3, 5, 2, 2, 3, 7, 4, 1, 2, 3, 4, 2, 5, 4, 2, 0, 4, 6, 2, 2, 2, 4, 3, 4

Offset: 1

Author

Wesley Ivan Hurt, Jan 08 2017

Keywords

• nonn
• easy
• changed

Examples

			a(20) = 2; there are 2 partitions of 20 into two squarefree semiprimes: (14,6) and (10,10).
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A006881, A071068, A280710.

Programs

Maple

with(numtheory): A280829:=n->add(floor(bigomega(i)*mobius(i)^2/2)*floor(2*mobius(i)^2/bigomega(i))*floor(bigomega(n-i)*mobius(i)^2/2)*floor(2*mobius(n-i)^2/bigomega(n-i)), i=2..floor(n/2)): seq(A280829(n), n=1..100);

Mathematica

Table[Sum[Floor[PrimeOmega[i] MoebiusMu[i]^2 / 2] Floor[2 MoebiusMu[i]^2 / PrimeOmega[i]] Floor[PrimeOmega[n - i] MoebiusMu[i]^2 / 2] Floor[2 MoebiusMu[n - i]^2 / PrimeOmega[n - i]], {i, 2, Floor[n/2]}], {n, 1, 90}] (* Indranil Ghosh, Mar 10 2017, translated from Maple code *)

PARI

for(n=1, 90, print1(sum(i=2, floor(n/2), floor(bigomega(i) * moebius(i)^2 / 2) * floor(2 * moebius(i)^2 / bigomega(i)) * floor(bigomega(n - i) * moebius(i)^2 / 2) * floor(2 * moebius(n - i)^2 / bigomega(n - i))),", ")) \\ Indranil Ghosh, Mar 10 2017

Formula

a(n) = Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A280710.