A331388 a(n) = Sum_{k=1..n} mu(gcd(n, k)) * k / gcd(n, k).

1, 0, 2, 3, 9, 3, 20, 12, 24, 10, 54, 15, 77, 21, 48, 48, 135, 24, 170, 57, 103, 55, 252, 60, 240, 78, 216, 123, 405, 47, 464, 192, 273, 136, 390, 144, 665, 171, 388, 228, 819, 102, 902, 327, 456, 253, 1080, 240, 1008, 240, 678, 465, 1377, 216, 1036, 492, 853, 406, 1710

Offset

1,3

Comments

Moebius transform of A023896.

Links

Table of n, a(n) for n=1..59.

Formula

a(n) = (1/n) * Sum_{k=1..n} mu(gcd(n, k)) * lcm(n, k).

a(n) = Sum_{d|n} mu(n/d) * A023896(d).

a(n) = Sum_{d|n} A007427(n/d) * A057661(d).

Sum_{k=1..n} a(k) ~ n^3 / (Pi^2 * Zeta(3)). - Vaclav Kotesovec, Feb 19 2020

Mathematica

Table[Sum[MoebiusMu[GCD[n, k]] k/GCD[n, k], {k, 1, n}], {n, 1, 65}]
A023896[n_] := Sum[If[GCD[n, k] == 1, k, 0], {k, 1, n}]; Table[Sum[MoebiusMu[n/d] A023896[d], {d, Divisors[n]}], {n, 1, 65}]

Prog

(Magma) [&+[MoebiusMu(Gcd(n, k))*(k div Gcd(n, k)):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Feb 18 2020

Crossrefs

Cf. A007427, A007431, A008683, A023896, A057661, A063445.

Sequence in context: A021422 A193086 A281289 * A093319 A160388 A358057

Adjacent sequences: A331385 A331386 A331387 * A331389 A331390 A331391

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 18 2020

Status

approved