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Numbers k such that the k-th cyclotomic polynomial has a root mod 13.
7

%I #22 Feb 09 2024 12:28:39

%S 1,2,3,4,6,12,13,26,39,52,78,156,169,338,507,676,1014,2028,2197,4394,

%T 6591,8788,13182,26364,28561,57122,85683,114244,171366,342732,371293,

%U 742586,1113879,1485172,2227758,4455516,4826809,9653618,14480427,19307236,28960854

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

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

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

%H Eric M. Schmidt, <a href="/A245481/b245481.txt">Table of n, a(n) for n = 1..500</a>

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,13).

%F 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

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

%t LinearRecurrence[{0,0,0,0,0,13},{1,2,3,4,6,12},50] (* _Harvey P. Dale_, Aug 19 2021 *)

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

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

%o (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

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

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

%K nonn,easy

%O 1,2

%A _Eric M. Schmidt_, Jul 23 2014