A345128 Number of squarefree products s*t from all positive integer pairs (s,t), such that s + t = n, s <= t.

0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 3, 4, 3, 3, 3, 2, 4, 4, 4, 4, 4, 4, 3, 5, 4, 5, 5, 4, 6, 4, 5, 7, 6, 5, 6, 8, 5, 9, 7, 6, 7, 8, 7, 8, 7, 5, 8, 10, 7, 6, 8, 7, 10, 9, 7, 11, 10, 8, 10, 8, 8, 11, 10, 9, 10, 11, 10, 13, 13, 9, 12, 10, 10, 14, 12, 12, 12, 13, 11, 12

Offset

1,7

Links

Table of n, a(n) for n=1..85.

Formula

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

Example

a(13) = 3; The partitions of 13 into two positive integer parts (s,t) where s <= t are (1,12), (2,11), (3,10), (4,9), (5,8), (6,7). The corresponding products are 1*12, 2*11, 3*10, 4*9, 5*8, and 6*7; 3 of which are squarefree.

Mathematica

Table[Sum[MoebiusMu[k (n - k)]^2, {k, Floor[n/2]}], {n, 100}]

Crossrefs

Cf. A008683 (mu).

Sequence in context: A269982 A329462 A153864 * A112399 A349040 A165123

Adjacent sequences: A345125 A345126 A345127 * A345129 A345130 A345131

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 08 2021

Status

approved