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A193679
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Sequence related to discriminant of cyclotomic polynomials A004124.
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5
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1, 2, 3, 4, 5, 12, 7, 16, 27, 80, 11, 144, 13, 448, 2025, 256, 17, 1728, 19, 6400, 35721, 11264, 23, 20736, 3125, 53248, 19683, 200704, 29, 518400, 31, 65536, 7144929, 1114112, 37515625, 2985984, 37, 4980736, 89813529, 40960000, 41, 146313216, 43, 126877696
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OFFSET
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1,2
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COMMENTS
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a(p) = p for primes p.
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REFERENCES
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P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, 2001, p. 297, eq.(1).
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LINKS
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Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
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FORMULA
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a(n) = n^phi(n)/abs(discriminant(Phi(n,x))), n>=1, with the cyclotomic polynomials Phi(n,x) and the Euler totient function phi(n)=A000010(n).
a(n) = product(p^(phi(n)/(p-1)),p prime dividing n), n>=2, a(1)=1.
Conjecture: Dirichlet g.f. of log(a(n)): -zeta(s-1)*zeta'(s)/zeta(s)^2, where zeta'(s) is the derivative of zeta(s). This would give a(n) = exp(Sum_{d|n} Lambda(d)*phi(n/d)), with Lambda(n)=log(A014963) and phi(n)=A000010. - Benedict W. J. Irwin, Jul 14 2018
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EXAMPLE
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n=6: a(6) = 2^(2/(2-1))*3^(2/(3-1)) = 12.
Discriminant(Phi(6,x)) = -3 = - (6^phi(6))/a(6).
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MAPLE
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MATHEMATICA
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a[n_] := n^EulerPhi[n]/Abs[Discriminant[Cyclotomic[n, x], x]]; Array[a, 44]
Table[Product[d^(-n*MoebiusMu[d]/d), {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, May 12 2024 *)
Table[Product[p^(EulerPhi[n]/(p-1)), {p, Select[Divisors[n], PrimeQ[#]&]}], {n, 1, 50}] (* Vaclav Kotesovec, May 13 2024 *)
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PROG
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(PARI) a(n) = n^eulerphi(n)/abs(poldisc(polcyclo(n))); \\ Michel Marcus, Jul 14 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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