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

%I A245481 #23 Feb 16 2025 08:33:23
%S A245481 1,2,3,4,6,12,13,26,39,52,78,156,169,338,507,676,1014,2028,2197,4394,
%T A245481 6591,8788,13182,26364,28561,57122,85683,114244,171366,342732,371293,
%U A245481 742586,1113879,1485172,2227758,4455516,4826809,9653618,14480427,19307236,28960854
%N A245481 Numbers k such that the k-th cyclotomic polynomial has a root mod 13.
%C A245481 Numbers of the form d*13^j for d a divisor of 12.
%D A245481 Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.
%H A245481 Eric M. Schmidt, <a href="/A245481/b245481.txt">Table of n, a(n) for n = 1..500</a>
%H A245481 Eric Weisstein, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%H A245481 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,13).
%F A245481 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 A245481 The 4th cyclotomic polynomial x^2 + 1 considered modulo 13 has a root x = 5, so 4 is in the sequence.
%t A245481 LinearRecurrence[{0,0,0,0,0,13},{1,2,3,4,6,12},50] (* _Harvey P. Dale_, Aug 19 2021 *)
%o A245481 (Sage) def A245481(n) : return [12,1,2,3,4,6][n%6]*13^((n-1)//6)
%o A245481 (PARI) for(n=1,10^6,if(#polrootsmod(polcyclo(n),13),print1(n,", "))) /* by definition; rather inefficient. - _Joerg Arndt_, Jul 28 2014 */
%o A245481 (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 A245481 (PARI) a(n)=[12,1,2,3,4,6][n%6+1]*13^((n-1)\6) \\ _Charles R Greathouse IV_, Jan 12 2017
%Y A245481 Cf. A018309, A038754, A245478, A245479, A245480.
%K A245481 nonn,easy
%O A245481 1,2
%A A245481 _Eric M. Schmidt_, Jul 23 2014