A069093 Jordan function J_8(n).

1, 255, 6560, 65280, 390624, 1672800, 5764800, 16711680, 43040160, 99609120, 214358880, 428236800, 815730720, 1470024000, 2562493440, 4278190080, 6975757440, 10975240800, 16983563040, 25499934720, 37817088000

Offset: 1

Author

Benoit Cloitre, Apr 05 2002

Keywords

• easy
• nonn
• mult

Comments

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Michael Lugo, A little number theory problem (2008)
• Wikipedia, Jordan's totient function.

Crossrefs

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)

Cf. A013667.

Programs

Maple

with(numtheory): seq(add(d^8 * mobius(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 13 2024

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 *)

PARI

for(n=1,100,print1(sumdiv(n,d,d^8*moebius(n/d)),","))

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)

Limit_{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)