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A059377 Jordan function J_4(n).

%I A059377 #68 Jan 30 2024 11:30:36
%S A059377 1,15,80,240,624,1200,2400,3840,6480,9360,14640,19200,28560,36000,
%T A059377 49920,61440,83520,97200,130320,149760,192000,219600,279840,307200,
%U A059377 390000,428400,524880,576000,707280,748800,923520,983040,1171200,1252800,1497600,1555200,1874160
%N A059377 Jordan function J_4(n).
%C A059377 This sequence is multiplicative. - _Mitch Harris_, Apr 19 2005
%C A059377 For n = 4 or n >= 6, a(n) is divisible by 240. - _Jianing Song_, Apr 06 2019
%D A059377 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%D A059377 R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.
%H A059377 T. D. Noe, <a href="/A059377/b059377.txt">Table of n, a(n) for n=1..1000</a>
%H A059377 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 A059377 Michael Lugo, <a href="http://godplaysdice.blogspot.com/2008/05/little-number-theory-problem.html">A little number theory problem</a> (2008)
%H A059377 László Tóth, <a href="http://arxiv.org/abs/1310.7053">Multiplicative arithmetic functions of several variables: a survey</a>, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
%H A059377 Wikipedia, <a href="https://en.wikipedia.org/wiki/Jordan%27s_totient_function">Jordan's totient function</a>.
%F A059377 a(n) = Sum_{d|n} d^4*mu(n/d). - _Benoit Cloitre_, Apr 05 2002
%F A059377 Multiplicative with a(p^e) = p^(4e)-p^(4(e-1)).
%F A059377 Dirichlet generating function: zeta(s-4)/zeta(s). - _Franklin T. Adams-Watters_, Sep 11 2005
%F A059377 a(n) = Sum_{k=1..n} gcd(k,n)^4 * cos(2*Pi*k/n). - _Enrique Pérez Herrero_, Jan 18 2013
%F A059377 a(n) = n^4*Product_{distinct primes p dividing n} (1 - 1/p^4). - _Tom Edgar_, Jan 09 2015
%F A059377 G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. - _Ilya Gutkovskiy_, Apr 25 2017
%F A059377 Sum_{k=1..n} a(k) ~ n^5 / (5*zeta(5)). - _Vaclav Kotesovec_, Feb 07 2019
%F A059377 From _Amiram Eldar_, Oct 12 2020: (Start)
%F A059377 lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^4 = 1/zeta(5).
%F A059377 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^4/(p^4-1)^2) = 1.0870036174... (End)
%F A059377 O.g.f.: Sum_{n >= 1} mu(n)*x^n*(1 + 11*x^n + 11*x^(2*n) + x^(3*n))/(1 - x^n)^5 = x + 15*x^2 + 80*x^3 + 240*x^4 + 624*x^5 + .... - _Peter Bala_, Jan 31 2022
%F A059377 From _Peter Bala_, Jan 01 2024: (Start)
%F A059377 a(n) = Sum_{d divides n} d * J_3(d) * J_1(n/d) = Sum_{d divides n} d^2 * J_2(d) * J_2(n/d) = Sum_{d divides n} d^3 * J_1(d) * J_3(n/d), where J_1(n) = phi(n) = A000010(n), J_2(n) = A007434(n) and J(3,n) = A059376(n).
%F A059377 a(n) = Sum_{k = 1..n} gcd(k, n) * J_3(gcd(k, n)) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_2(gcd(j, k, n)) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n)^3 * J_1(gcd(i, j, k, n)). (End)
%F A059377 a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} J_2(i) * J_2(j) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i) * J_3(j) (apply Lehmer, Theorem 1). - _Peter Bala_, Jan 29 2024
%p A059377 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 A059377 seq(J(n,4), n=1..40);
%t A059377 JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 4]; Array[f, 38]
%t A059377 f[p_, e_] := p^(4*e) - p^(4*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 12 2020 *)
%o A059377 (PARI) for(n=1,100,print1(sumdiv(n,d,d^4*moebius(n/d)),","))
%o A059377 (PARI) a(n)=if(n<1,0,sumdiv(n,d,d^4*moebius(n/d)))
%o A059377 (PARI) a(n)=if(n<1,0,dirdiv(vector(n,k,k^4),vector(n,k,1))[n])
%o A059377 (PARI) { for (n = 1, 1000, write("b059377.txt", n, " ", sumdiv(n, d, d^4*moebius(n/d))); ) } \\ _Harry J. Smith_, Jun 26 2009
%Y A059377 See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059378 (J_5), A069091 - A069095 (J_6 through J_10).
%Y A059377 Cf. A013663.
%K A059377 nonn,mult,easy
%O A059377 1,2
%A A059377 _N. J. A. Sloane_, Jan 28 2001