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A280829 Number of partitions of n into two squarefree semiprimes.

%I A280829 #21 Aug 31 2025 16:47:28
%S A280829 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,
%T A280829 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,
%U A280829 2,2,3,7,4,1,2,3,4,2,5,4,2,0,4,6,2,2,2,4,3,4
%N A280829 Number of partitions of n into two squarefree semiprimes.
%H A280829 Indranil Ghosh, <a href="/A280829/b280829.txt">Table of n, a(n) for n = 1..1000</a>
%H A280829 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A280829 a(n) = Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A280710.
%e A280829 a(20) = 2; there are 2 partitions of 20 into two squarefree semiprimes: (14,6) and (10,10).
%p A280829 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);
%t A280829 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 *)
%o A280829 (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
%Y A280829 Cf. A006881, A071068, A280710.
%K A280829 nonn,easy,changed
%O A280829 1,20
%A A280829 _Wesley Ivan Hurt_, Jan 08 2017