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

1,12

Links

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A280710, A280829.

Sequence in context: A271437 A271517 A200512 * A104203 A242240 A225341

Adjacent sequences: A280829 A280830 A280831 * A280833 A280834 A280835

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 08 2017

Status

approved