A302391 Number of partitions of 2n into two parts with at least one nonsquarefree part.

0, 0, 1, 1, 3, 2, 4, 3, 5, 4, 6, 5, 8, 7, 10, 8, 10, 7, 12, 9, 13, 9, 15, 11, 17, 12, 17, 15, 18, 15, 20, 17, 22, 16, 22, 15, 23, 18, 26, 20, 28, 20, 29, 21, 28, 23, 30, 24, 32, 25, 33, 26, 34, 23, 36, 27, 37, 27, 39, 29, 41, 29, 40, 34, 42, 34, 44, 36, 46

Offset: 1

Author

Wesley Ivan Hurt, Apr 06 2018

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A013929, A280226, A294097.

Programs

Magma

[&+[(1-MoebiusMu(2*n-k)^2*MoebiusMu(k)^2): k in [1..n]]: n in [1..70]]; // Vincenzo Librandi, Apr 09 2018

Mathematica

Table[Sum[1 - MoebiusMu[2 n - i]^2*MoebiusMu[i]^2, {i, n}], {n, 100}]

PARI

a(n) = sum(i=1, n, 1 - moebius(2*n-i)^2*moebius(i)^2); \\ Michel Marcus, Apr 09 2018

Formula

a(n) = Sum_{i=1..n} 1 - mu(2n-i)^2 * mu(i)^2, where mu is the Möbius function (A008683).

a(n) = n - A280226(n). - Wesley Ivan Hurt, Dec 11 2023