|
%I #88 Jan 30 2024 11:41:52
%S 1,7,26,56,124,182,342,448,702,868,1330,1456,2196,2394,3224,3584,4912,
%T 4914,6858,6944,8892,9310,12166,11648,15500,15372,18954,19152,24388,
%U 22568,29790,28672,34580,34384,42408,39312,50652,48006,57096
%N Jordan function J_3(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="/A059376/b059376.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 Multiplicative with a(p^e) = p^(3e) - p^(3e-3). - _Vladeta Jovovic_, Jul 26 2001
%F a(n) = Sum_{d|n} d^3*mu(n/d). - _Benoit Cloitre_, Apr 05 2002
%F Dirichlet generating function: zeta(s-3)/zeta(s). - _Franklin T. Adams-Watters_, Sep 11 2005
%F A063453(n) divides a(n). - _R. J. Mathar_, Mar 30 2011
%F a(n) = Sum_{k=1..n} gcd(k,n)^3 * cos(2*Pi*k/n). - _Enrique Pérez Herrero_, Jan 18 2013
%F a(n) = n^3*Product_{distinct primes p dividing n} (1-1/p^3). - _Tom Edgar_, Jan 09 2015
%F 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
%F Sum_{d|n} a(d) = n^3. - _Werner Schulte_, Jan 12 2018
%F Sum_{k=1..n} a(k) ~ 45*n^4 / (2*Pi^4). - _Vaclav Kotesovec_, Feb 07 2019
%F From _Amiram Eldar_, Oct 12 2020: (Start)
%F lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^3 = 1/zeta(4) (A215267).
%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^3/(p^3-1)^2) = 1.2253556451... (End)
%F 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
%F From _Peter Bala_, Jan 01 2024
%F 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).
%F 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)
%F 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
%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 = 3)
%p A059376 := proc(n)
%p add(d^3*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
%p end proc: # _R. J. Mathar_, Nov 03 2015
%t JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 3]; Array[f, 39]
%t 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 *)
%o (PARI) for(n=1,120,print1(sumdiv(n,d,d^3*moebius(n/d)),","))
%o (PARI) for (n = 1, 1000, write("b059376.txt", n, " ", sumdiv(n, d, d^3*moebius(n/d))); ) \\ _Harry J. Smith_, Jun 26 2009
%o (PARI) seq(n) = dirmul(vector(n,k,k^3), vector(n,k,moebius(k)));
%o seq(39) \\ _Gheorghe Coserea_, May 11 2016
%o (Python)
%o from math import prod
%o from sympy import factorint
%o 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
%Y 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).
%Y Cf. A013662, A215267.
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_, Jan 28 2001
|
