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

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, 30, 31, 64, 0, 0, 35, 108, 37, 0, 39, 80, 82, 42, 86, 132, 90, 0, 94, 192, 147, 50, 0, 156, 106, 108, 110, 168, 114, 0, 118, 240, 305, 0, 63, 128, 195, 132, 201, 340, 138, 140

Offset: 1

Author

Wesley Ivan Hurt, Jan 08 2017

Keywords

• nonn
• easy
• changed

Examples

			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.
		

Links

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

Crossrefs

Cf. A280710, A280829.

Programs

Maple

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);

Mathematica

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

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

Formula

a(n) = n * Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A280710. - Wesley Ivan Hurt, Aug 31 2025