A245481 Numbers k such that the k-th cyclotomic polynomial has a root mod 13.

1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156, 169, 338, 507, 676, 1014, 2028, 2197, 4394, 6591, 8788, 13182, 26364, 28561, 57122, 85683, 114244, 171366, 342732, 371293, 742586, 1113879, 1485172, 2227758, 4455516, 4826809, 9653618, 14480427, 19307236, 28960854

Offset: 1

Author

Eric M. Schmidt, Jul 23 2014

Keywords

• nonn
• easy

Comments

Numbers of the form d*13^j for d a divisor of 12.

Examples

			The 4th cyclotomic polynomial x^2 + 1 considered modulo 13 has a root x = 5, so 4 is in the sequence.
		

References

• Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..500
• Eric Weisstein, Cyclotomic Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,13).

Crossrefs

Cf. A018309, A038754, A245478, A245479, A245480.

Programs

Mathematica

LinearRecurrence[{0,0,0,0,0,13},{1,2,3,4,6,12},50] (* Harvey P. Dale, Aug 19 2021 *)

PARI

for(n=1,10^6,if(#polrootsmod(polcyclo(n),13),print1(n,", "))) /* by definition; rather inefficient. - Joerg Arndt, Jul 28 2014 */

PARI

Vec(-x*(12*x^5+6*x^4+4*x^3+3*x^2+2*x+1)/(13*x^6-1) + O(x^100)) \\ Colin Barker, Jul 30 2014

PARI

a(n)=[12,1,2,3,4,6][n%6+1]*13^((n-1)\6) \\ Charles R Greathouse IV, Jan 12 2017

Sage

def A245481(n) : return [12,1,2,3,4,6][n%6]*13^((n-1)//6)

Formula

a(n) = 13*a(n-6). G.f.: -x*(12*x^5+6*x^4+4*x^3+3*x^2+2*x+1) / (13*x^6-1). - Colin Barker, Jul 30 2014