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A069091 Jordan function J_6(n). 9


%S 1,63,728,4032,15624,45864,117648,258048,530712,984312,1771560,

%T 2935296,4826808,7411824,11374272,16515072,24137568,33434856,47045880,

%U 62995968,85647744,111608280,148035888,187858944,244125000,304088904,386889048

%N Jordan function J_6(n).

%C From _Enrique Pérez Herrero_, Sep 14 2010: (Start)

%C a(n) is the Moebius transform of n^6.

%C Note that J_2(n), J_3(n), eulerphi(n) and psi(n) divides a(n), this sequences

%C are: A007434(n), A059376(n), A000010(n) and A001615(n) respectively. (End)

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

%H Enrique Pérez Herrero, <a href="/A069091/b069091.txt">Table of n, a(n) for n=1..2000</a>

%F a(n) = Sum_{d|n} d^6*mu(n/d).

%F Multiplicative with a(p^e) = p^(6e)-p^(6(e-1)).

%F Dirichlet generating function: zeta(s-6)/zeta(s). - _Ralf Stephan_, Jul 04 2013

%F a(n) = n^6*Product_{distinct primes p dividing n} (1-1/p^6). - _Tom Edgar_, Jan 09 2015

%F Sum_{k=1..n} a(k) ~ n^7 / (7*Zeta(7)). - _Vaclav Kotesovec_, Feb 07 2019

%t JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]

%t A069091[n_IntegerQ]:=JordanTotient[n,6]; (* _Enrique Pérez Herrero_, Sep 14 2010 *)

%o (PARI) for(n=1,100,print1(sumdiv(n,d,d^6*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. A065959. [_Enrique Pérez Herrero_, Sep 14 2010]

%K easy,nonn,mult

%O 1,2

%A _Benoit Cloitre_, Apr 05 2002

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)