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A020500 Cyclotomic polynomials at x=1.

%I A020500 #57 Aug 25 2024 09:27:13
%S A020500 0,2,3,2,5,1,7,2,3,1,11,1,13,1,1,2,17,1,19,1,1,1,23,1,5,1,3,1,29,1,31,
%T A020500 2,1,1,1,1,37,1,1,1,41,1,43,1,1,1,47,1,7,1,1,1,53,1,1,1,1,1,59,1,61,1,
%U A020500 1,2,1,1,67,1,1,1,71,1,73,1,1,1,1,1,79,1,3
%N A020500 Cyclotomic polynomials at x=1.
%C A020500 Also the greatest common divisor of the prime factors of n. - _Peter Luschny_, Mar 22 2011
%H A020500 T. D. Noe, <a href="/A020500/b020500.txt">Table of n, a(n) for n = 1..10000</a>
%H A020500 Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/papers/cyclotomic.html">Cyclotomic polynomials and prime numbers</a>
%H A020500 Bartlomiej Bzdega, Andres Herrera-Poyatos, Pieter Moree, <a href="https://arxiv.org/abs/1611.06783">Cyclotomic polynomials at roots of unity</a>, arXiv:1611.06783 [math.NT], 2016. See Lemma 19.
%H A020500 <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index entries for cyclotomic polynomials, values at X</a>
%F A020500 a(1) = 0; for n > 1, a(n) = gcd(lpf(n),gpf(n)), by Gallot's theorem 1.4. - _Thomas Ordowski_, May 04 2013
%F A020500 For n > 2, a(n) = lcm(1,2,...,n)/lcm(1,...,n-1). - _Thomas Ordowski_, Nov 01 2013
%p A020500 with(numtheory,cyclotomic); f := n->subs(x=1,cyclotomic(n,x)); seq(f(i),i=0..64);
%p A020500 A020500 := n -> igcd(op(numtheory[factorset](n))):
%p A020500 seq(A020500(i), i=1..73); # _Peter Luschny_, Mar 22 2011
%t A020500 Table[ Cyclotomic[n, 1], {n, 1, 73}] (* _Jean-François Alcover_, Jan 10 2013 *)
%t A020500 Join[{0},Table[GCD@@FactorInteger[n][[All,1]],{n,2,80}]] (* _Harvey P. Dale_, Jul 18 2019 *)
%o A020500 (PARI) a(n) = polcyclo(n, 1); \\ _Michel Marcus_, Oct 23 2015
%o A020500 (PARI) a(n) = if (n==1, 0, if (isprimepower(n,&p), p, 1)); \\ _Michel Marcus_, Nov 23 2016
%Y A020500 Apart from initial zero, same as A014963.
%Y A020500 Cf. A007947.
%K A020500 nonn,easy,nice
%O A020500 1,2
%A A020500 _Simon Plouffe_