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A280832 Sum of the parts in the partitions of n into two squarefree semiprimes.

%I A280832 #25 Aug 31 2025 16:43:13
%S A280832 0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,16,0,0,0,40,21,0,0,24,25,0,27,56,29,
%T A280832 30,31,64,0,0,35,108,37,0,39,80,82,42,86,132,90,0,94,192,147,50,0,156,
%U A280832 106,108,110,168,114,0,118,240,305,0,63,128,195,132,201,340,138,140
%N A280832 Sum of the parts in the partitions of n into two squarefree semiprimes.
%H A280832 Indranil Ghosh, <a href="/A280832/b280832.txt">Table of n, a(n) for n = 1..1000</a>
%H A280832 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A280832 a(n) = n * Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A280710. - _Wesley Ivan Hurt_, Aug 31 2025
%e A280832 a(20) = 40; there are two partitions of n into two squarefree semiprimes: (14,6) and (10,10). The sum of the parts in these partitions is 40.
%p A280832 with(numtheory): A280832:=n->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(A280832(n), n=1..100);
%t A280832 Table[n 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, 70}] (* _Indranil Ghosh_, Mar 09 2017, translated from Maple code *)
%t A280832 spp[n_]:=Total[Flatten[Select[IntegerPartitions[n,{2}],AllTrue[#,SquareFreeQ] && PrimeOmega[ #]=={2,2}&]]]; Array[spp,70] (* _Harvey P. Dale_, Jun 12 2022 *)
%o A280832 (PARI) for(n=1, 70, print1(n * 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 09 2017, translated from Maple code
%Y A280832 Cf. A280710, A280829.
%K A280832 nonn,easy,changed
%O A280832 1,12
%A A280832 _Wesley Ivan Hurt_, Jan 08 2017