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

1,2

Comments

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

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).

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A218337 A335993 A018309 * A039038 A034293 A136291

Adjacent sequences: A245478 A245479 A245480 * A245482 A245483 A245484

Keyword

nonn,easy

Author

Eric M. Schmidt, Jul 23 2014

Status

approved