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Sum of the parts in the partitions of n into two squarefree semiprimes.
1

%I #20 Jun 12 2022 13:15:17

%S 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 30,31,64,0,0,35,108,37,0,39,80,82,42,86,132,90,0,94,192,147,50,0,156,

%U 106,108,110,168,114,0,118,240,305,0,63,128,195,132,201,340,138,140

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

%H Indranil Ghosh, <a href="/A280832/b280832.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%e 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 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 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 spp[n_]:=Total[Flatten[Select[IntegerPartitions[n,{2}],AllTrue[#,SquareFreeQ] && PrimeOmega[ #]=={2,2}&]]]; Array[spp,70] (* _Harvey P. Dale_, Jun 12 2022 *)

%o (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 Cf. A280829.

%K nonn,easy

%O 1,12

%A _Wesley Ivan Hurt_, Jan 08 2017