OFFSET
1,2
COMMENTS
This sequence is multiplicative. - Mitch Harris, Apr 19 2005
For n = 4 or n >= 6, a(n) is divisible by 240. - Jianing Song, Apr 06 2019
REFERENCES
L. 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
T. D. Noe, Table of n, a(n) for n=1..1000
D. H. Lehmer, On a theorem of von Sterneck, Bull. Amer. Math. Soc. 37(10): 723-726 (1931)
Michael Lugo, A little number theory problem (2008)
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
Wikipedia, Jordan's totient function.
FORMULA
a(n) = Sum_{d|n} d^4*mu(n/d). - Benoit Cloitre, Apr 05 2002
Multiplicative with a(p^e) = p^(4e)-p^(4(e-1)).
Dirichlet generating function: zeta(s-4)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = Sum_{k=1..n} gcd(k,n)^4 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
a(n) = n^4*Product_{distinct primes p dividing n} (1 - 1/p^4). - Tom Edgar, Jan 09 2015
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
Sum_{k=1..n} a(k) ~ n^5 / (5*zeta(5)). - Vaclav Kotesovec, Feb 07 2019
From Amiram Eldar, Oct 12 2020: (Start)
lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^4 = 1/zeta(5).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^4/(p^4-1)^2) = 1.0870036174... (End)
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
From Peter Bala, Jan 01 2024: (Start)
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).
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)
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
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:
seq(J(n, 4), n=1..40);
MATHEMATICA
JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 4]; Array[f, 38]
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 *)
PROG
(PARI) for(n=1, 100, print1(sumdiv(n, d, d^4*moebius(n/d)), ", "))
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, d^4*moebius(n/d)))
(PARI) a(n)=if(n<1, 0, dirdiv(vector(n, k, k^4), vector(n, k, 1))[n])
(PARI) { for (n = 1, 1000, write("b059377.txt", n, " ", sumdiv(n, d, d^4*moebius(n/d))); ) } \\ Harry J. Smith, Jun 26 2009
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
N. J. A. Sloane, Jan 28 2001
STATUS
approved