This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193679 #31 Feb 16 2025 08:33:15
%S A193679 1,2,3,4,5,12,7,16,27,80,11,144,13,448,2025,256,17,1728,19,6400,35721,
%T A193679 11264,23,20736,3125,53248,19683,200704,29,518400,31,65536,7144929,
%U A193679 1114112,37515625,2985984,37,4980736,89813529,40960000,41,146313216,43,126877696
%N A193679 Sequence related to discriminant of cyclotomic polynomials A004124.
%C A193679 a(p) = p for primes p.
%D A193679 P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, 2001, p. 297, eq.(1).
%H A193679 Alois P. Heinz, <a href="/A193679/b193679.txt">Table of n, a(n) for n = 1..1000</a>
%H A193679 Mohammad K. Azarian, <a href="http://www.ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
%H A193679 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>
%F A193679 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).
%F A193679 a(n) = product(p^(phi(n)/(p-1)),p prime dividing n), n>=2, a(1)=1.
%F A193679 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
%e A193679 n=6: a(6) = 2^(2/(2-1))*3^(2/(3-1)) = 12.
%e A193679 Discriminant(Phi(6,x)) = -3 = - (6^phi(6))/a(6).
%p A193679 with(numtheory): A193679 := n -> n^phi(n)/abs(discrim(cyclotomic (n,x),x)); seq(A193679(i),i=1..49); # _Peter Luschny_, Aug 20 2011
%t A193679 a[n_] := n^EulerPhi[n]/Abs[Discriminant[Cyclotomic[n, x], x]]; Array[a, 44]
%t A193679 (* _Jean-François Alcover_, Mar 21 2017 *)
%t A193679 Table[Product[d^(-n*MoebiusMu[d]/d), {d, Divisors[n]}], {n, 1, 50}] (* _Vaclav Kotesovec_, May 12 2024 *)
%t A193679 Table[Product[p^(EulerPhi[n]/(p-1)), {p, Select[Divisors[n], PrimeQ[#]&]}], {n, 1, 50}] (* _Vaclav Kotesovec_, May 13 2024 *)
%o A193679 (PARI) a(n) = n^eulerphi(n)/abs(poldisc(polcyclo(n))); \\ _Michel Marcus_, Jul 14 2018
%Y A193679 Cf. A004124.
%K A193679 nonn
%O A193679 1,2
%A A193679 _Wolfdieter Lang_, Aug 20 2011