A345129 Sum of the squarefree products s*t from all positive integer pairs (s,t), such that s + t = n, s <= t.

0, 1, 2, 3, 6, 5, 16, 22, 14, 21, 40, 46, 94, 46, 40, 109, 208, 159, 182, 161, 148, 268, 296, 380, 380, 472, 488, 497, 770, 620, 666, 851, 740, 1082, 560, 1015, 1506, 1226, 946, 1490, 2088, 1381, 2566, 1941, 2160, 2379, 2832, 2489, 2976, 3111, 2290, 3832, 4732, 3395, 3340

Offset

1,3

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

a(13) = 94; 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. The sum of the squarefree products from this list is 22 + 30 + 42 = 94.

Mathematica

Table[Sum[k (n - k) MoebiusMu[k (n - k)]^2, {k, Floor[n/2]}], {n, 80}]

Crossrefs

Cf. A008683 (mu), A325148.

Sequence in context: A222311 A156833 A367370 * A329354 A332461 A319344

Adjacent sequences: A345126 A345127 A345128 * A345130 A345131 A345132

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 08 2021

Status

approved