A059376 Jordan function J_3(n).

1, 7, 26, 56, 124, 182, 342, 448, 702, 868, 1330, 1456, 2196, 2394, 3224, 3584, 4912, 4914, 6858, 6944, 8892, 9310, 12166, 11648, 15500, 15372, 18954, 19152, 24388, 22568, 29790, 28672, 34580, 34384, 42408, 39312, 50652, 48006, 57096

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), A007434 (J_2), A059377 (J_4), A059378 (J_5), A069091 - A069095 (J_6 through J_10).

Cf. A013662, A215267.

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 = 3)
A059376 := proc(n)
    add(d^3*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Nov 03 2015

Mathematica

JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 3]; Array[f, 39]
f[p_, e_] := p^(3*e) - p^(3*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2020 *)

PARI

for(n=1,120,print1(sumdiv(n,d,d^3*moebius(n/d)),","))

PARI

for (n = 1, 1000, write("b059376.txt", n, " ", sumdiv(n, d, d^3*moebius(n/d))); ) \\ Harry J. Smith, Jun 26 2009

PARI

seq(n) = dirmul(vector(n,k,k^3), vector(n,k,moebius(k)));
seq(39)  \\ Gheorghe Coserea, May 11 2016

Python

from math import prod
from sympy import factorint
def A059376(n): return prod(p**(3*(e-1))*(p**3-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

Formula

Multiplicative with a(p^e) = p^(3e) - p^(3e-3). - Vladeta Jovovic, Jul 26 2001

a(n) = Sum_{d|n} d^3*mu(n/d). - Benoit Cloitre, Apr 05 2002

Dirichlet generating function: zeta(s-3)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005

A063453(n) divides a(n). - R. J. Mathar, Mar 30 2011

a(n) = Sum_{k=1..n} gcd(k,n)^3 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013

a(n) = n^3*Product_{distinct primes p dividing n} (1-1/p^3). - Tom Edgar, Jan 09 2015

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

Sum_{d|n} a(d) = n^3. - Werner Schulte, Jan 12 2018

Sum_{k=1..n} a(k) ~ 45*n^4 / (2*Pi^4). - Vaclav Kotesovec, Feb 07 2019

From Amiram Eldar, Oct 12 2020: (Start)

lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^3 = 1/zeta(4) (A215267).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^3/(p^3-1)^2) = 1.2253556451... (End)

O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 4*x^n + x^(2*n))/(1 - x^n)^4 = x + 7*x^2 + 26*x^3 + 56*x^4 + 124*x^5 + .... - Peter Bala, Jan 31 2022

From Peter Bala, Jan 01 2024: (Start)

a(n) = Sum_{d divides n} d * J_2(d) * phi(n/d) = Sum_{d divides n} d^2 * phi(d) * J_2(n/d), where J_2(n) = A007434(n).

a(n) = Sum_{k = 1..n} gcd(k, n) * J_2(gcd(k, n)) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_1(gcd(j, k, n)). (End)

a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i)*J_2(j) = Sum_{1 <= i, j, k <= n, lcm(i, j, k) = n} phi(i)*phi(j)*phi(k), where J_2(n) = A007434(n). - Peter Bala, Jan 29 2024