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A059377
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Jordan function J_4(n).
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25
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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
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OFFSET
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1,2
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COMMENTS
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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
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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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T. D. Noe, Table of n, a(n) for n=1..1000
Wikipedia, Jordan's totient function.
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FORMULA
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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)
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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 = 4)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059376 (J_3), A059378 (J_5).
Cf. A013663.
Sequence in context: A085808 A180577 A033594 * A123865 A024002 A050149
Adjacent sequences: A059374 A059375 A059376 * A059378 A059379 A059380
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KEYWORD
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nonn,mult,easy
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AUTHOR
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N. J. A. Sloane, Jan 28 2001
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STATUS
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approved
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