A069095 Jordan function J_10(n).

1, 1023, 59048, 1047552, 9765624, 60406104, 282475248, 1072693248, 3486725352, 9990233352, 25937424600, 61855850496, 137858491848, 288972178704, 576640565952, 1098437885952, 2015993900448, 3566920035096, 6131066257800

Offset: 1

Author

Benoit Cloitre, Apr 05 2002

Keywords

• easy
• nonn
• mult

Comments

a(n) is divisible by 264 = (2^3)*3*11 = A006863(5), except for n = 1, 2, 3 or 11. See Lugo. - Peter Bala, Jan 13 2024

References

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

Links

• Robert Israel, 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 - A069094 (J_6 through J_9).

Cf. A013669.

Programs

Maple

f:= n -> n^10*mul(1-1/p^10, p=numtheory:-factorset(n)):
map(f, [$1..30]); # Robert Israel, Jan 09 2015

Mathematica

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 10]; Array[f, 21]
f[p_, e_] := p^(10*e) - p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PARI

a(n) = sumdiv(n,d,d^10*moebius(n/d));

Formula

a(n) = Sum_{d|n} d^10*mu(n/d).

Multiplicative with a(p^e) = p^(10e)-p^(10(e-1)).

Dirichlet generating function: zeta(s-10)/zeta(s). - Ralf Stephan, Jul 04 2013

a(n) = n^10*Product_{distinct primes p dividing n} (1-1/p^10). - Tom Edgar, Jan 09 2015

Sum_{k=1..n} a(k) ~ n^11 / (11*zeta(11)). - Vaclav Kotesovec, Feb 07 2019

From Amiram Eldar, Oct 12 2020: (Start)

lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^10 = 1/zeta(11).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^10/(p^10-1)^2) = 1.0009955309... (End)