A332617 a(n) = Sum_{k=1..n} J_n(k), where J is the Jordan function, J_n(k) = k^n * Product_{p|k, p prime} (1 - 1/p^n).

1, 4, 34, 336, 4390, 66312, 1197858, 24612000, 574002448, 14903406552, 427622607366, 13419501812640, 457579466056498, 16840326075104280, 665473192580864556, 28101209228393371200, 1262896789586657015796, 60182268296582518426368, 3031282541337682050032664

Offset: 1

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

• nonn

Links

• Wikipedia, Jordan's totient function

Crossrefs

Cf. A000010, A002088, A007434, A059376, A059377, A059378, A059379, A059380, A067858, A319194, A321879.

Programs

Magma

[&+[&+[MoebiusMu(k div d)*d^n:d in Divisors(k)]:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020

Mathematica

Table[Sum[Sum[MoebiusMu[k/d] d^n, {d, Divisors[k]}], {k, 1, n}], {n, 1, 19}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[Sum[MoebiusMu[k] j^n x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} Sum_{j>=1} mu(k) * j^n * x^(k*j).