A020513 - OEIS

%I #33 Aug 26 2024 13:33:59

%S -1,-2,0,1,2,1,3,1,2,1,5,1,1,1,7,1,2,1,3,1,1,1,11,1,1,1,13,1,1,1,1,1,

%T 2,1,17,1,1,1,19,1,1,1,1,1,1,1,23,1,1,1,5,1,1,1,3,1,1,1,29,1,1,1,31,1,

%U 2

%N Cyclotomic polynomials evaluated at x=-1.

%H Robert Israel, <a href="/A020513/b020513.txt">Table of n, a(n) for n = 0..10000</a>

%H E. T. Bell, <a href="http://dx.doi.org/10.1090/S0002-9904-1927-04394-4">Cauchy’s cyclotomic function and functional powers</a>, Bull. Amer. Math. Soc. 33 (1927), 416-422.

%H Bartlomiej Bzdega, Andres Herrera-Poyatos, and Pieter Moree, <a href="https://arxiv.org/abs/1611.06783">Cyclotomic polynomials at roots of unity</a>, arXiv:1611.06783 [math.NT], 2016-2017. See Lemma 7.

%F For n >= 3: if n = 2*p^m with a prime p then a(n) = p otherwise a(n) = 1. - Ola Veshta (olaveshta(AT)my-deja.com), Jun 01 2001

%p with(numtheory,cyclotomic); f := n->subs(x=-1,cyclotomic(n,x)); seq(f(i),i=0..64);

%t Array[Cyclotomic[#, -1] &, 90, 0] (* _Robert G. Wilson v_, Nov 23 2016 *)

%o (PARI) a(n) = if (n==0, -1, subst(polcyclo(n), x, -1)); \\ _Michel Marcus_, Apr 22 2016

%o (PARI) a(n) = if (n==0, -1, if (n==1, -2, if (n==2, 0, if (!(n % 2) && isprimepower(n/2,&p), p, 1)))); \\ _Michel Marcus_, Nov 23 2016

%o (Python)

%o from sympy import primefactors

%o def A020513(n): return (-1,-2,0)[n] if n<3 else (f[0] if n&1^1 and len(f:=primefactors(n>>1))==1 else 1) # _Chai Wah Wu_, Aug 26 2024

%Y Cf. A138929 (2*p^m, m >= 0 values).

%K sign

%O 0,2

%A _Simon Plouffe_