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A059378
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Jordan function J_5(n).
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22
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1, 31, 242, 992, 3124, 7502, 16806, 31744, 58806, 96844, 161050, 240064, 371292, 520986, 756008, 1015808, 1419856, 1822986, 2476098, 3099008, 4067052, 4992550, 6436342, 7682048, 9762500, 11510052, 14289858, 16671552, 20511148
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OFFSET
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1,2
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REFERENCES
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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.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^(5e)-p^(5(e-1)).
a(n) = n^5*Product_{distinct primes p dividing n} (1-1/p^5). - Tom Edgar, Jan 09 2015
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
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^5 = 1/zeta(6).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^5/(p^5-1)^2) = 1.0379908060... (End)
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
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).
a(n) = Sum_{k = 1..n} gcd(k, n) * J_4(gcd(k, n)).
a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)^2 * J_3(gcd(j, k, n)). (End)
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
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MAPLE
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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)
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MATHEMATICA
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JordanJ[n_, k_] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 5]; Array[f, 30]
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 *)
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PROG
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(PARI) for(n=1, 100, print1(sumdiv(n, d, d^5*moebius(n/d)), ", "))
(PARI) { for (n = 1, 1000, write("b059378.txt", n, " ", sumdiv(n, d, d^5*moebius(n/d))); ) } \\ Harry J. Smith, Jun 26 2009
(Python)
from sympy import divisors, mobius
def a(n):
return sum(d**5 * mobius(n // d) for d in divisors(n))
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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