A069094 Jordan function J_9(n).

1, 511, 19682, 261632, 1953124, 10057502, 40353606, 133955584, 387400806, 998046364, 2357947690, 5149441024, 10604499372, 20620692666, 38441386568, 68585259008, 118587876496, 197961811866, 322687697778, 510999738368

Offset: 1

Author

Benoit Cloitre, Apr 05 2002

Keywords

• easy
• nonn
• mult

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Wikipedia, Jordan's totient function.

Crossrefs

Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).

Cf. A013668.

Programs

Mathematica

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 9]; Array[f, 22]
f[p_, e_] := p^(9*e) - p^(9*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PARI

for(n=1,100,print1(sumdiv(n,d,d^9*moebius(n/d)),","))

Formula

a(n) = Sum_{d|n} d^9*mu(n/d).

Multiplicative with a(p^e) = p^(9e)-p^(9(e-1)).

Dirichlet generating function: zeta(s-9)/zeta(s). - Ralf Stephan, Jul 04 2013

a(n) = n^9*Product_{distinct primes p dividing n} (1-1/p^9). - Tom Edgar, Jan 09 2015

Sum_{k=1..n} a(k) ~ 18711*n^10 / (2*Pi^10). - Vaclav Kotesovec, Feb 07 2019

From Amiram Eldar, Oct 12 2020: (Start)

lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^9 = 1/zeta(10).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^9/(p^9-1)^2) = 1.0020122252... (End)