A059378 - OEIS

%I #60 Jan 30 2024 11:41:57

%S 1,31,242,992,3124,7502,16806,31744,58806,96844,161050,240064,371292,

%T 520986,756008,1015808,1419856,1822986,2476098,3099008,4067052,

%U 4992550,6436342,7682048,9762500,11510052,14289858,16671552,20511148

%N Jordan function J_5(n).

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

%D R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

%H T. D. Noe, <a href="/A059378/b059378.txt">Table of n, a(n) for n=1..1000</a>

%H D. H. Lehmer, <a href="https://doi.org/10.1090/S0002-9904-1931-05247-2">On a theorem of von Sterneck</a>, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>.

%F a(n) = Sum_{d|n} d^5*mu(n/d). - _Benoit Cloitre_, Apr 05 2002

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

%F Dirichlet generating function: zeta(s-5)/zeta(s). - _Franklin T. Adams-Watters_, Sep 11 2005.

%F a(n) = n^5*Product_{distinct primes p dividing n} (1-1/p^5). - _Tom Edgar_, Jan 09 2015

%F 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

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

%F From _Amiram Eldar_, Oct 12 2020: (Start)

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

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

%F 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

%F From _Peter Bala_, Jan 01 2024

%F 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).

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

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

%F 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

%p 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)

%t JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 5]; Array[f, 30]

%t 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 *)

%o (PARI) for(n=1,100,print1(sumdiv(n,d,d^5*moebius(n/d)),","))

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

%o (Python)

%o from sympy import divisors, mobius

%o def a(n):

%o     return sum(d**5 * mobius(n // d) for d in divisors(n))

%o # _Indranil Ghosh_, Apr 26 2017

%Y 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).

%Y Cf. A013664.

%K nonn,mult,easy

%O 1,2

%A _N. J. A. Sloane_, Jan 28 2001