A020500 Cyclotomic polynomials at x=1.

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, 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, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3

Offset: 1

Author

Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

Also the greatest common divisor of the prime factors of n. - Peter Luschny, Mar 22 2011

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Yves Gallot, Cyclotomic polynomials and prime numbers
• Bartlomiej Bzdega, Andres Herrera-Poyatos, Pieter Moree, Cyclotomic polynomials at roots of unity, arXiv:1611.06783 [math.NT], 2016. See Lemma 19.
• Index entries for cyclotomic polynomials, values at X

Crossrefs

Apart from initial zero, same as A014963.

Cf. A007947.

Programs

Maple

with(numtheory,cyclotomic); f := n->subs(x=1,cyclotomic(n,x)); seq(f(i),i=0..64);
A020500 := n -> igcd(op(numtheory[factorset](n))):
seq(A020500(i), i=1..73); # Peter Luschny, Mar 22 2011

Mathematica

Table[ Cyclotomic[n, 1], {n, 1, 73}] (* Jean-François Alcover, Jan 10 2013 *)
Join[{0},Table[GCD@@FactorInteger[n][[All,1]],{n,2,80}]] (* Harvey P. Dale, Jul 18 2019 *)

PARI

a(n) = polcyclo(n, 1); \\ Michel Marcus, Oct 23 2015

PARI

a(n) = if (n==1, 0, if (isprimepower(n,&p), p, 1)); \\ Michel Marcus, Nov 23 2016

Formula

a(1) = 0; for n > 1, a(n) = gcd(lpf(n),gpf(n)), by Gallot's theorem 1.4. - Thomas Ordowski, May 04 2013

For n > 2, a(n) = lcm(1,2,...,n)/lcm(1,...,n-1). - Thomas Ordowski, Nov 01 2013