This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A245479 #27 Feb 16 2025 08:33:23
%S A245479 1,2,3,6,7,14,21,42,49,98,147,294,343,686,1029,2058,2401,4802,7203,
%T A245479 14406,16807,33614,50421,100842,117649,235298,352947,705894,823543,
%U A245479 1647086,2470629,4941258,5764801,11529602,17294403,34588806,40353607,80707214,121060821
%N A245479 Numbers n such that the n-th cyclotomic polynomial has a root mod 7.
%C A245479 Numbers of the form d*7^j for d = 1,2,3,6.
%D A245479 Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.
%H A245479 Eric M. Schmidt, <a href="/A245479/b245479.txt">Table of n, a(n) for n = 1..500</a>
%H A245479 Benedict W. J. Irwin <a href="https://www.authorea.com/users/5445/articles/119155/_show_article">A sequence a(n+1) is product over binary components of n, plus 1</a>
%H A245479 Eric Weisstein, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%H A245479 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,7).
%F A245479 a(n) = 7*a(n-4). G.f.: -x*(2*x+1)*(3*x^2+1) / (7*x^4-1). - _Colin Barker_, Jul 30 2014
%F A245479 From _Benedict W. J. Irwin_, Jul 22 2016: (Start)
%F A245479 a(n) appears to satisfy x*Prod_{n>=0} (1 + a(2^n+1)x^(2^n)) = Sum_{n>=1} a(n)x^n.
%F A245479 Then a(n+1)=a(2^x+1)a(2^y+1)a(2^z+1)..., where n=2^x+2^y+2^z+... .
%F A245479 For example,
%F A245479 n=12=2^2+2^3, then a(12+1)=a(2^2+1)*a(2^3+1) i.e. 343=49*7.
%F A245479 n=31=2^0+2^1+2^2+2^3+2^4, then a(31+1)=a(2)*a(3)*a(5)*a(9)*a(17) i.e. 4941258=2*3*7*49*2401.
%F A245479 (End)
%e A245479 The 3rd cyclotomic polynomial x^2 + x + 1 considered modulo 7 has a root x = 2, so 3 is in the sequence.
%t A245479 m = 7; Function[d, Table[d[[k]] m^n, {n, 0, 9}, {k, Length@ d}]]@ Divisors[m - 1] // Flatten (* or *)
%t A245479 Rest@ CoefficientList[Series[-x (2 x + 1) (3 x^2 + 1)/(7 x^4 - 1), {x, 0, 40}], x] (* _Michael De Vlieger_, Jul 25 2016 *)
%t A245479 LinearRecurrence[{0,0,0,7},{1,2,3,6},50] (* _Harvey P. Dale_, Oct 10 2018 *)
%o A245479 (Sage) def A245479(n) : return [6,1,2,3][n%4]*7^((n-1)//4)
%o A245479 (PARI) for(n=1,10^6,if(#polrootsmod(polcyclo(n),7),print1(n,", "))) /* by definition; rather inefficient. - _Joerg Arndt_, Jul 28 2014 */
%o A245479 (PARI) Vec(-x*(2*x+1)*(3*x^2+1)/(7*x^4-1) + O(x^100)) \\ _Colin Barker_, Jul 30 2014
%Y A245479 Cf. A018379, A038754, A245478, A245480, A245481.
%K A245479 nonn,easy
%O A245479 1,2
%A A245479 _Eric M. Schmidt_, Jul 23 2014