A093106 Numbers k such that the k-th cyclotomic polynomial evaluated at 2 (=A019320(k)) is not coprime to k.

6, 18, 20, 21, 54, 100, 110, 136, 147, 155, 156, 162, 253, 342, 486, 500, 602, 657, 812, 820, 889, 979, 1029, 1081, 1210, 1332, 1458, 2028, 2265, 2312, 2485, 2500, 2756, 3081, 3164, 3422, 3660, 3924, 4112, 4374, 4422, 4656, 4805, 5253, 5784, 5819, 6498

Offset: 1

Author

Ralf Stephan, Mar 20 2004

Keywords

• nonn

Comments

Also, numbers k such that the Zsigmondy number Zs(k, 2, 1) differs from the k-th cyclotomic polynomial evaluated at 2, i.e., A064078(k) differs from A019320(k).

Numbers k > 0 such that A019320(k) is not congruent to 1 mod k. These numbers are of the form k = p^j * A002326((p-1)/2), where p is an odd prime and j > 0. Then A019320(k) mod k = gcd(A019320(k), k) = A019320(k) / A064078(k) = p. - Thomas Ordowski, Oct 07 2017

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 1..51350

Crossrefs

Cf. A093107, A093108, A093109.

Programs

Mathematica

Select[Range[10000],GCD[#,Cyclotomic[#,2]]!=1 &] (* Emmanuel Vantieghem, Nov 13 2016 *)

PARI

isok(k) = gcd(polcyclo(k, 2), k) != 1; \\ Michel Marcus, Oct 07 2017

PARI

upto(K)=li=List();forprime(p=3,K*log(2)/log(K+1),r=znorder(Mod(2,p))*p;while(r<=K,listput(li,r);r*=p));Set(li) \\ Jeppe Stig Nielsen, Sep 10 2020

Extensions

More terms from Vladeta Jovovic, Apr 03 2004

Definition corrected by Jerry Metzger, Nov 04 2009

Edited by Max Alekseyev, Oct 23 2017