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
Keywords
Programs
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Magma
[&+[MoebiusMu(Gcd(n,k))*(k div Gcd(n,k)):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Feb 18 2020
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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}]
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).
Sum_{k=1..n} a(k) ~ n^3 / (Pi^2 * Zeta(3)). - Vaclav Kotesovec, Feb 19 2020
Comments