A303051 Number of partitions of n into two distinct parts (p,q) such that p, q and p+q are all squarefree.
0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 2, 0, 3, 2, 3, 0, 4, 0, 3, 0, 4, 4, 4, 0, 0, 4, 0, 0, 5, 4, 5, 0, 6, 6, 6, 0, 7, 6, 7, 0, 8, 7, 9, 0, 0, 7, 8, 0, 0, 0, 7, 0, 10, 0, 7, 0, 10, 10, 9, 0, 11, 10, 0, 0, 11, 10, 11, 0, 12, 12, 11, 0, 13, 13, 0, 0, 14, 12, 14, 0, 0
Offset: 1
Programs
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Mathematica
Table[Sum[MoebiusMu[n]^2 MoebiusMu[i]^2 MoebiusMu[n - i]^2, {i, Floor[(n - 1)/2]}], {n, 100}] -
PARI
a(n) = sum(i=1, (n-1)\2, moebius(n)^2*moebius(i)^2*moebius(n-i)^2); \\ Michel Marcus, Apr 17 2018
Formula
a(n) = Sum_{i=1..floor((n-1)/2)} mu(n)^2 * mu(i)^2 * mu(n-i)^2, where mu is the Möbius function (A008683).