A059377 - OEIS (original) (raw)

1, 15, 80, 240, 624, 1200, 2400, 3840, 6480, 9360, 14640, 19200, 28560, 36000, 49920, 61440, 83520, 97200, 130320, 149760, 192000, 219600, 279840, 307200, 390000, 428400, 524880, 576000, 707280, 748800, 923520, 983040, 1171200, 1252800, 1497600, 1555200, 1874160

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.

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)).

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

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

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

For all n > 1, k*n^4 < a(n) < n^4 for k = 1/zeta(4) = 90/Pi^4 = 0.9239.... - Charles R Greathouse IV, Sep 29 2025

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