A020513 Cyclotomic polynomials evaluated at x=-1.

-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, 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, 2

Offset: 0

Author

Simon Plouffe

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• E. T. Bell, Cauchy’s cyclotomic function and functional powers, Bull. Amer. Math. Soc. 33 (1927), 416-422.
• Bartlomiej Bzdega, Andres Herrera-Poyatos, and Pieter Moree, Cyclotomic polynomials at roots of unity, arXiv:1611.06783 [math.NT], 2016-2017. See Lemma 7.

Crossrefs

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

Programs

Maple

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

Mathematica

Array[Cyclotomic[#, -1] &, 90, 0] (* Robert G. Wilson v, Nov 23 2016 *)

PARI

a(n) = if (n==0, -1, subst(polcyclo(n), x, -1)); \\ Michel Marcus, Apr 22 2016

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

Python

from sympy import primefactors
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

Formula

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