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

%I A059379 #37 Jun 08 2025 05:01:34
%S A059379 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,
%T A059379 1,4,48,182,624,992,728,127,1,6,48,342,1200,3124,4032,2186,255,1,4,72,
%U A059379 448,2400,7502,15624,16256,6560,511,1,10,72,702,3840
%N A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).
%D A059379 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%D A059379 R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.
%H A059379 Enrique Pérez Herrero, <a href="/A059379/b059379.txt">Table of n, a(n) for n = 1..10000</a>
%F A059379 J_k(n) = Sum_{d|n} d^k*mu(n/d). - _Benoit Cloitre_ and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002
%F A059379 From _Amiram Eldar_, Jun 07 2025: (Start)
%F A059379 For a given k, J_k(n) is multiplicative with J_k(p^e) = p^(k*e) - p^(k*e-k).
%F A059379 For a given k, Dirichlet g.f. of J_k(n): zeta(s-k)/zeta(s).
%F A059379 Sum_{i=1..n} J_k(i) ~ n^(k+1) / ((k+1)*zeta(k+1)).
%F A059379 Sum_{n>=1} 1/J_k(n) = Product_{p prime} (1 + p^k/(p^k-1)^2) for k >= 2. (End)
%e A059379 Array begins:
%e A059379   1,  1,  2,   2,   4,    2,    6,    4,   6,  4, 10, 4, ...
%e A059379   1,  3,  8,  12,  24,   24,   48,   48,  72, 72, ...
%e A059379   1,  7, 26,  56, 124,  182,  342,  448, 702, ...
%e A059379   1, 15, 80, 240, 624, 1200, 2400, 3840, ...
%p A059379 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;
%p A059379 #alternative
%p A059379 A059379 := proc(n,k)
%p A059379     add(d^k*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
%p A059379 end proc:
%p A059379 seq(seq(A059379(d-k,k),k=1..d-1),d=2..12) ; # _R. J. Mathar_, Nov 23 2018
%t A059379 JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
%t A059379 A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
%t A059379 A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
%t A059379 A059379[n_]:=JordanTotient[A004736[n],A002260[n]]; (* _Enrique Pérez Herrero_, Dec 19 2010 *)
%o A059379 (PARI)
%o A059379 jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
%o A059379 A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
%o A059379 A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
%o A059379 A059379(n)=jordantot(A004736(n),A002260(n)); \\ _Enrique Pérez Herrero_, Jan 08 2011
%o A059379 (Python)
%o A059379 from functools import cache
%o A059379 def MoebiusTrans(a, i):
%o A059379     @cache
%o A059379     def mb(n, d = 1):
%o A059379           return d % n and -mb(d, n % d < 1) + mb(n, d + 1) or 1 // n
%o A059379     def mob(m, n): return mb(m // n) if m % n == 0 else 0
%o A059379     return sum(mob(i, d) * a(d) for d in range(1, i + 1))
%o A059379 def Jrow(n, size):
%o A059379     return [MoebiusTrans(lambda m: m ** n, k) for k in range(1, size)]
%o A059379 for n in range(1, 8): print(Jrow(n, 13))
%o A059379 # Alternatively:
%o A059379 from sympy import primefactors as prime_divisors
%o A059379 from fractions import Fraction as QQ
%o A059379 from math import prod as product
%o A059379 def J(n: int, k: int) -> int:
%o A059379     t = QQ(pow(k, n), 1)
%o A059379     s = product(1 - QQ(1, pow(p, n)) for p in prime_divisors(k))
%o A059379     return (t * s).numerator  # the denominator is always 1
%o A059379 for n in range(1, 8): print([J(n, k) for k in range(1, 13)])
%o A059379 # _Peter Luschny_, Dec 16 2023
%Y A059379 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.
%Y A059379 Main diagonal gives A067858.
%K A059379 nonn,tabl
%O A059379 1,4
%A A059379 _N. J. A. Sloane_, Jan 28 2001