OFFSET
1,2
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
LINKS
Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
Wikipedia, Jordan's totient function.
FORMULA
a(n) = Sum_{d|n} d^7*mu(n/d).
Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).
Dirichlet generating function: zeta(s-7)/zeta(s). - Ralf Stephan, Jul 04 2013
a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - Tom Edgar, Jan 09 2015
Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)
O.g.f.: Sum_{n >= 1} mu(n)*A_7(x^n)/(1 - x^n)^8 = x + 127*x^2 + 2186*x^3 + 16256*x^4 + 78124*x^5 + ..., where A_7(x) = x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7 is the 7th Eulerian polynomial. See A008292. - Peter Bala, Jan 31 2022
MATHEMATICA
JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
A069092[n_] := JordanTotient[n, 7]; (* Enrique Pérez Herrero, Nov 02 2010 *)
f[p_, e_] := p^(7*e) - p^(7*(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^7*moebius(n/d)), ", "))
CROSSREFS
KEYWORD
easy,nonn,mult
AUTHOR
Benoit Cloitre, Apr 05 2002
STATUS
approved