A302986 Number of partitions of n into two distinct parts (p,q) such that p, q and |q-p| are all squarefree.
0, 0, 1, 1, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 3, 3, 4, 0, 2, 3, 3, 0, 4, 4, 5, 0, 4, 3, 4, 0, 4, 5, 5, 0, 4, 7, 5, 0, 6, 6, 7, 0, 8, 7, 9, 0, 6, 7, 8, 0, 5, 7, 7, 0, 6, 6, 8, 0, 8, 7, 9, 0, 11, 7, 9, 0, 8, 10, 8, 0, 10, 13, 12, 0, 10, 11, 11, 0, 11, 11, 15, 0, 9
Offset: 1
Programs
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Magma
[0,0] cat [&+[(MoebiusMu(k)^2*MoebiusMu(n-k)^2)*MoebiusMu(n-2*k)^2: k in [1..Floor((n-1)/2)]]: n in [3..100]]; // Vincenzo Librandi, Apr 17 2018
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Mathematica
Table[Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2*MoebiusMu[n - 2 i]^2, {i, Floor[(n - 1)/2]}], {n, 100}] Table[Count[IntegerPartitions[n,{2}],?(AllTrue[{#[[1]],#[[2]],#[[1]] - #[[2]]},SquareFreeQ]&)],{n,90}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jan 21 2021 *) -
PARI
a(n) = sum(i=1, (n-1)\2, moebius(i)^2*moebius(n-i)^2*moebius(n-2*i)^2); \\ Michel Marcus, Apr 17 2018
Formula
a(n) = Sum_{i=1..floor((n-1)/2)} mu(i)^2 * mu(n-i)^2 * mu(n-2*i)^2, where mu is the Möbius function (A008683).
a(n) = 0 for n in A111284. - Michel Marcus, Apr 17 2018