A069091 Jordan function J_6(n).

1, 63, 728, 4032, 15624, 45864, 117648, 258048, 530712, 984312, 1771560, 2935296, 4826808, 7411824, 11374272, 16515072, 24137568, 33434856, 47045880, 62995968, 85647744, 111608280, 148035888, 187858944, 244125000, 304088904, 386889048

Offset: 1

Author

Benoit Cloitre, Apr 05 2002

Keywords

• easy
• nonn
• mult

Comments

Moebius transform of n^6. - Enrique Pérez Herrero, Sep 14 2010

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

References

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

Links

• Enrique Pérez Herrero, Table of n, a(n) for n=1..2000
• 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), A069092 - A069095 (J_7 through J_10).

Cf. A065959.

Cf. A013665.

Programs

Maple

with(numtheory): seq(add(d^6 * mobius(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 13 2024

Mathematica

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]
A069091[n_IntegerQ]:=JordanTotient[n,6]; (* Enrique Pérez Herrero, Sep 14 2010 *)
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 *)

PARI

for(n=1,100,print1(sumdiv(n,d,d^6*moebius(n/d)),","))

Formula

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

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

Dirichlet generating function: zeta(s-6)/zeta(s). - Ralf Stephan, Jul 04 2013

a(n) = n^6*Product_{distinct primes p dividing n} (1-1/p^6). - Tom Edgar, Jan 09 2015

Sum_{k=1..n} a(k) ~ n^7 / (7*zeta(7)). - Vaclav Kotesovec, Feb 07 2019

From Amiram Eldar, Oct 12 2020: (Start)

Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^6 = 1/zeta(7).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^6/(p^6-1)^2) = 1.0175973008... (End)

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