A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

1, 1, 1, 2, 3, 1, 2, 8, 7, 1, 4, 12, 26, 15, 1, 2, 24, 56, 80, 31, 1, 6, 24, 124, 240, 242, 63, 1, 4, 48, 182, 624, 992, 728, 127, 1, 6, 48, 342, 1200, 3124, 4032, 2186, 255, 1, 4, 72, 448, 2400, 7502, 15624, 16256, 6560, 511, 1, 10, 72, 702, 3840

Offset: 1

Author

N. J. A. Sloane, Jan 28 2001

Keywords

• nonn
• tabl

Examples

			Array begins:
  1,  1,  2,   2,   4,    2,    6,    4,   6,  4, 10, 4, ...
  1,  3,  8,  12,  24,   24,   48,   48,  72, 72, ...
  1,  7, 26,  56, 124,  182,  342,  448, 702, ...
  1, 15, 80, 240, 624, 1200, 2400, 3840, ...
		

References

• Louis 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

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Crossrefs

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.

Main diagonal gives A067858.

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;
#alternative
A059379 := proc(n,k)
    add(d^k*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc:
seq(seq(A059379(d-k,k),k=1..d-1),d=2..12) ; # R. J. Mathar, Nov 23 2018

Mathematica

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059379[n_]:=JordanTotient[A004736[n],A002260[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)

PARI

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059379(n)=jordantot(A004736(n),A002260(n)); \\ Enrique Pérez Herrero, Jan 08 2011

Python

from functools import cache
def MoebiusTrans(a, i):
    @cache
    def mb(n, d = 1):
          return d % n and -mb(d, n % d < 1) + mb(n, d + 1) or 1 // n
    def mob(m, n): return mb(m // n) if m % n == 0 else 0
    return sum(mob(i, d) * a(d) for d in range(1, i + 1))
def Jrow(n, size):
    return [MoebiusTrans(lambda m: m ** n, k) for k in range(1, size)]
for n in range(1, 8): print(Jrow(n, 13))
# Alternatively:
from sympy import primefactors as prime_divisors
from fractions import Fraction as QQ
from math import prod as product
def J(n: int, k: int) -> int:
    t = QQ(pow(k, n), 1)
    s = product(1 - QQ(1, pow(p, n)) for p in prime_divisors(k))
    return (t * s).numerator  # the denominator is always 1
for n in range(1, 8): print([J(n, k) for k in range(1, 13)])
# Peter Luschny, Dec 16 2023

Formula

J_k(n) = Sum_{d|n} d^k*mu(n/d). - Benoit Cloitre and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002

From Amiram Eldar, Jun 07 2025: (Start)

For a given k, J_k(n) is multiplicative with J_k(p^e) = p^(k*e) - p^(k*e-k).

For a given k, Dirichlet g.f. of J_k(n): zeta(s-k)/zeta(s).

Sum_{i=1..n} J_k(i) ~ n^(k+1) / ((k+1)*zeta(k+1)).

Sum_{n>=1} 1/J_k(n) = Product_{p prime} (1 + p^k/(p^k-1)^2) for k >= 2. (End)