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
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
Programs
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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
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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 *)
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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
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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)