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 A069093 Jordan function J_8(n). 6
 1, 255, 6560, 65280, 390624, 1672800, 5764800, 16711680, 43040160, 99609120, 214358880, 428236800, 815730720, 1470024000, 2562493440, 4278190080, 6975757440, 10975240800, 16983563040, 25499934720, 37817088000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3. LINKS G. C. Greubel, Table of n, a(n) for n = 1..10000 Wikipedia, Jordan's totient function. FORMULA a(n) = Sum_{d|n} d^8*mu(n/d). Multiplicative with a(p^e) = p^(8e)-p^(8(e-1)). Dirichlet generating function: zeta(s-8)/zeta(s). - Ralf Stephan, Jul 04 2013 a(n) = n^8*Product_{distinct primes p dividing n} (1-1/p^8). - Tom Edgar, Jan 09 2015 Sum_{k=1..n} a(k) ~ n^9 / (9*zeta(9)). - Vaclav Kotesovec, Feb 07 2019 From Amiram Eldar, Oct 12 2020: (Start) lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^8 = 1/zeta(9). Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^8/(p^8-1)^2) = 1.0040927606... (End) MATHEMATICA JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 8]; Array[f, 25] 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 *) PROG (PARI) for(n=1, 100, print1(sumdiv(n, d, d^8*moebius(n/d)), ", ")) CROSSREFS Cf. A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Cf. A013667. Sequence in context: A228223 A022524 A261032 * A024006 A258809 A321553 Adjacent sequences:  A069090 A069091 A069092 * A069094 A069095 A069096 KEYWORD easy,nonn,mult AUTHOR Benoit Cloitre, Apr 05 2002 STATUS approved

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Last modified January 23 08:11 EST 2021. Contains 340385 sequences. (Running on oeis4.)