OFFSET
1,2
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.
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)
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 *)
PROG
(PARI) for(n=1, 100, print1(sumdiv(n, d, d^9*moebius(n/d)), ", "))
CROSSREFS
KEYWORD
easy,nonn,mult
AUTHOR
Benoit Cloitre, Apr 05 2002
STATUS
approved