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%I #38 Feb 02 2022 23:39:13
%S 1,127,2186,16256,78124,277622,823542,2080768,4780782,9921748,
%T 19487170,35535616,62748516,104589834,170779064,266338304,410338672,
%U 607159314,893871738,1269983744,1800262812,2474870590,3404825446,4548558848
%N Jordan function J_7(n).
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H Enrique Pérez Herrero, <a href="/A069092/b069092.txt">Table of n, a(n) for n=1..2000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>.
%F a(n) = Sum_{d|n} d^7*mu(n/d).
%F Multiplicative with a(p^e) = p^(7e)-p^(7(e-1)).
%F Dirichlet generating function: zeta(s-7)/zeta(s). - _Ralf Stephan_, Jul 04 2013
%F a(n) = n^7*Product_{distinct primes p dividing n} (1-1/p^7). - _Tom Edgar_, Jan 09 2015
%F Sum_{k=1..n} a(k) ~ 4725*n^8 / (4*Pi^8). - _Vaclav Kotesovec_, Feb 07 2019
%F From _Amiram Eldar_, Oct 12 2020: (Start)
%F lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^7 = 1/zeta(8).
%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^7/(p^7-1)^2) = 1.0084115178... (End)
%F 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
%t JordanTotient[n_, k_: 1] := DivisorSum[n, (#^k)*MoebiusMu[n/# ] &] /; (n > 0) && IntegerQ[n]
%t A069092[n_] := JordanTotient[n, 7]; (* _Enrique Pérez Herrero_, Nov 02 2010 *)
%t 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 *)
%o (PARI) for(n=1, 100, print1(sumdiv(n, d, d^7*moebius(n/d)), ", "))
%Y Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5).
%Y Cf. A069091 (J_6), A069092 (J_7), A069093 (J_8), A069094 (J_9), A069095 (J_10). [_Enrique Pérez Herrero_, Nov 02 2010]
%Y Cf. A013666.
%K easy,nonn,mult
%O 1,2
%A _Benoit Cloitre_, Apr 05 2002
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