This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A069093 #41 Jan 17 2024 09:08:02
%S A069093 1,255,6560,65280,390624,1672800,5764800,16711680,43040160,99609120,
%T A069093 214358880,428236800,815730720,1470024000,2562493440,4278190080,
%U A069093 6975757440,10975240800,16983563040,25499934720,37817088000
%N A069093 Jordan function J_8(n).
%C A069093 a(n) is divisible by 480 = (2^5)*3*5 = A006863(4), except for n = 1, 2, 3 and 5. See Lugo. - _Peter Bala_, Jan 13 2024
%D A069093 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H A069093 G. C. Greubel, <a href="/A069093/b069093.txt">Table of n, a(n) for n = 1..10000</a>
%H A069093 Michael Lugo, <a href="http://godplaysdice.blogspot.com/2008/05/little-number-theory-problem.html">A little number theory problem</a> (2008)
%H A069093 Wikipedia, <a href="https://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>.
%F A069093 a(n) = Sum_{d|n} d^8*mu(n/d).
%F A069093 Multiplicative with a(p^e) = p^(8e)-p^(8(e-1)).
%F A069093 Dirichlet generating function: zeta(s-8)/zeta(s). - _Ralf Stephan_, Jul 04 2013
%F A069093 a(n) = n^8*Product_{distinct primes p dividing n} (1-1/p^8). - _Tom Edgar_, Jan 09 2015
%F A069093 Sum_{k=1..n} a(k) ~ n^9 / (9*zeta(9)). - _Vaclav Kotesovec_, Feb 07 2019
%F A069093 From _Amiram Eldar_, Oct 12 2020: (Start)
%F A069093 Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^8 = 1/zeta(9).
%F A069093 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^8/(p^8-1)^2) = 1.0040927606... (End)
%p A069093 with(numtheory): seq(add(d^8 * mobius(n/d), d in divisors(n)), n = 1..100); # _Peter Bala_, Jan 13 2024
%t A069093 JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 8]; Array[f, 25]
%t A069093 f[p_, e_] := p^(8*e) - p^(8*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 12 2020 *)
%o A069093 (PARI) for(n=1,100,print1(sumdiv(n,d,d^8*moebius(n/d)),","))
%Y A069093 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), A069091 - A069095 (J_6 through J_10)
%Y A069093 Cf. A013667.
%K A069093 easy,nonn,mult
%O A069093 1,2
%A A069093 _Benoit Cloitre_, Apr 05 2002