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A069091 Jordan function J_6(n).

%I A069091 #53 Jan 17 2024 09:08:32
%S A069091 1,63,728,4032,15624,45864,117648,258048,530712,984312,1771560,
%T A069091 2935296,4826808,7411824,11374272,16515072,24137568,33434856,47045880,
%U A069091 62995968,85647744,111608280,148035888,187858944,244125000,304088904,386889048
%N A069091 Jordan function J_6(n).
%C A069091 Moebius transform of n^6. - _Enrique Pérez Herrero_, Sep 14 2010
%C A069091 a(n) is divisible by 504 = (2^3)*(3^3)*7 = A006863(3) except for n = 1, 2, 3 and 7. See Lugo. - _Peter Bala_, Jan 13 2024
%D A069091 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H A069091 Enrique Pérez Herrero, <a href="/A069091/b069091.txt">Table of n, a(n) for n=1..2000</a>
%H A069091 Michael Lugo, <a href="http://godplaysdice.blogspot.com/2008/05/little-number-theory-problem.html">A little number theory problem</a> (2008)
%H A069091 Wikipedia, <a href="https://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>.
%F A069091 a(n) = Sum_{d|n} d^6*mu(n/d).
%F A069091 Multiplicative with a(p^e) = p^(6e)-p^(6(e-1)).
%F A069091 Dirichlet generating function: zeta(s-6)/zeta(s). - _Ralf Stephan_, Jul 04 2013
%F A069091 a(n) = n^6*Product_{distinct primes p dividing n} (1-1/p^6). - _Tom Edgar_, Jan 09 2015
%F A069091 Sum_{k=1..n} a(k) ~ n^7 / (7*zeta(7)). - _Vaclav Kotesovec_, Feb 07 2019
%F A069091 From _Amiram Eldar_, Oct 12 2020: (Start)
%F A069091 Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^6 = 1/zeta(7).
%F A069091 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^6/(p^6-1)^2) = 1.0175973008... (End)
%F A069091 O.g.f.: Sum_{n >= 1} mu(n)*A(x^n)/(1 - x^n)^7 = x + 63*x^2 + 728*x^3 + 4032*x^4 + 15624*x^5 + ..., where A(x) = x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6 is the 6th Eulerian polynomial. See A008292. - _Peter Bala_, Jan 31 2022
%p A069091 with(numtheory): seq(add(d^6 * mobius(n/d), d in divisors(n)), n = 1..100); # _Peter Bala_, Jan 13 2024
%t A069091 JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]
%t A069091 A069091[n_IntegerQ]:=JordanTotient[n,6]; (* _Enrique Pérez Herrero_, Sep 14 2010 *)
%t A069091 f[p_, e_] := p^(6*e) - p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 12 2020 *)
%o A069091 (PARI) for(n=1,100,print1(sumdiv(n,d,d^6*moebius(n/d)),","))
%Y A069091 Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5), A069092 - A069095 (J_7 through J_10).
%Y A069091 Cf. A065959.
%Y A069091 Cf. A013665.
%K A069091 easy,nonn,mult
%O A069091 1,2
%A A069091 _Benoit Cloitre_, Apr 05 2002