A059378 Jordan function J_5(n).

1, 31, 242, 992, 3124, 7502, 16806, 31744, 58806, 96844, 161050, 240064, 371292, 520986, 756008, 1015808, 1419856, 1822986, 2476098, 3099008, 4067052, 4992550, 6436342, 7682048, 9762500, 11510052, 14289858, 16671552, 20511148, 23436248, 28629150, 32505856

Offset: 1

Author

N. J. A. Sloane, Jan 28 2001

Keywords

• nonn
• mult
• easy

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
• R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• D. H. Lehmer, On a theorem of von Sterneck, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)
• Wikipedia, Jordan's totient function.

Crossrefs

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A069091 - A069095 (J_6 through J_10).

Cf. A013664.

Programs

Maple

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 5)

Mathematica

JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 5]; Array[f, 30]
f[p_, e_] := p^(5*e) - p^(5*(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^5*moebius(n/d)),","))

PARI

{ for (n = 1, 1000, write("b059378.txt", n, " ", sumdiv(n, d, d^5*moebius(n/d))); ) } \\ Harry J. Smith, Jun 26 2009

Python

from sympy import divisors, mobius
def a(n):
    return sum(d**5 * mobius(n // d) for d in divisors(n))
# Indranil Ghosh, Apr 26 2017

Formula

a(n) = Sum_{d|n} d^5*mu(n/d). - Benoit Cloitre, Apr 05 2002

Multiplicative with a(p^e) = p^(5e)-p^(5(e-1)).

Dirichlet generating function: zeta(s-5)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005

a(n) = n^5*Product_{distinct primes p dividing n} (1-1/p^5). - Tom Edgar, Jan 09 2015

G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^6. - Ilya Gutkovskiy, Apr 25 2017

Sum_{k=1..n} a(k) ~ 315*n^6 / (2*Pi^6). - Vaclav Kotesovec, Feb 07 2019

From Amiram Eldar, Oct 12 2020: (Start)

Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^5 = 1/zeta(6).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^5/(p^5-1)^2) = 1.0379908060... (End)

O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 26*x^n + 66*x^(2*n) + 26*x^(3*n) + x^(4*n))/(1 - x^n)^6 = x + 31*x^2 + 242*x^3 + 992*x^4 + 3124*x^5 + .... - Peter Bala, Jan 31 2022

From Peter Bala, Jan 01 2024: (Start)

a(n) = Sum_{d divides n} d * J_4(d) * J_1(n/d) = Sum_{d divides n} d^2 * J_3(d) * J_2(n/d) = Sum_{d divides n} d^3 * J_2(d) * J_3(n/d) = Sum_{d divides n} d^4 * J_1(d) * J_4(n/d), where J_1(n) = phi(n) = A000010(n), J_2(n) = A007434(n), J(3,n) = A059376(n) and J_4(n) = A059377(n).

a(n) = Sum_{k = 1..n} gcd(k, n) * J_4(gcd(k, n)).

a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_3(gcd(j, k, n)). (End)

a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} J_2(i) * J_3(j) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i) * J_4(j) (apply Lehmer, Theorem 1). - Peter Bala, Jan 30 2024