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 A384490 #22 Jun 05 2025 23:17:05
%S A384490 41,61,109,149,241,269,389,409,449,569,601,641,701,821,929,1129,1181,
%T A384490 1201,1301,1321,1429,1481,1489,1609,1801,1889,1901,1949,2129,2141,
%U A384490 2309,2341,2381,2549,2609,2741,2909,3061,3109,3181,3209,3221,3229,3361,3449,3541
%N A384490 Numbers m such that both roots of x^2 - x - 1 modulo m are primitive roots modulo m.
%C A384490 Empirical observation: For each m in this sequence A001175(m) = m-1 and A015134(m) = m+2.
%H A384490 Wikipedia, <a href="https://en.wikipedia.org/wiki/Primitive_root_modulo_n">Primitive root modulo n</a>.
%e A384490 For m = 41 the roots of x^2 - x - 1 (mod 41) are 7 and 35. 7 and 35 are both primitive roots modulo 41.
%t A384490 test[p_]:=Module[{inv2,sqr},If[JacobiSymbol[5,p]==1,inv2=ModularInverse[2,p]; sqr=PowerMod[5,1/2,p]; {MultiplicativeOrder[Mod[inv2*(sqr-1),p],p],MultiplicativeOrder[Mod[inv2*(-sqr-1),p],p]} == {p-1,p-1},False]]; Cases[Prime[Range[4, 5000]], _?(test[#] &)] (* _Shenghui Yang_, Jun 01 2025 *)
%o A384490 (PARI) { forprime(p=2, 3600, s=polrootsmod(x^2 - x - 1,p);
%o A384490 if( #s==2 && p-1==znorder(Mod(s[1],p)) && p-1==znorder(Mod(s[2],p)),
%o A384490 print1(p,", "); ); ); } \\ _Joerg Arndt_, May 31 2025
%Y A384490 Cf. A001175, A015134.
%K A384490 nonn
%O A384490 1,1
%A A384490 _Jay Anderson_, May 31 2025