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

1,7

Links

Table of n, a(n) for n=1..83.

Index entries for sequences related to partitions

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A008683, A111284, A294247, A303051.

Sequence in context: A300075 A056557 A342624 * A295305 A082900 A171958

Adjacent sequences: A302983 A302984 A302985 * A302987 A302988 A302989

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 16 2018

Status

approved