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 A122869 #37 Jun 20 2025 20:21:58
%S A122869 11,19,31,59,71,79,131,139,151,179,191,199,211,239,251,271,311,331,
%T A122869 359,379,419,431,439,479,491,499,571,599,619,631,659,691,719,739,751,
%U A122869 811,839,859,911,919,971,991,1019,1031,1039,1051,1091,1151,1171,1231,1259
%N A122869 Primes p that divide Lucas((p-1)/2), where Lucas is A000032.
%C A122869 Final digit of a(n) is 1 or 9.
%C A122869 A002145 is the union of this sequence and A122870, Primes p that divide Lucas((p+1)/2).
%C A122869 Conjecture: This sequence is just the primes congruent to 11 or 19 mod 20. - _Charles R Greathouse IV_, May 25 2011 [The conjecture is correct. - _Jianing Song_, Jun 20 2025]
%C A122869 Note that F(p-1) = F((p-1)/2)*Lucas((p-1)/2), where F = A000045. Since gcd(F(n),Lucas(n)) = 1 or 2 (because Lucas(n)^2 - 5*F(n)^2 = 4*(-1)^n), this sequence lists primes p such that p divides F(p-1) but does not divides F((p-1)/2). By Propositions 1.1 and 1.2 (the k = 3 case) of my link below, this is primes p == 11, 19 (mod 20). - _Jianing Song_, Jun 20 2025
%H A122869 Amiram Eldar, <a href="/A122869/b122869.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)
%H A122869 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%H A122869 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian Prime</a>.
%H A122869 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%t A122869 Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1-1)/2-1]+Fibonacci[(#1-1)/2+1])/#1]&]
%o A122869 (PARI) lista(kmax) = {my(lucas1 = 1, lucas2 = 3, lucas3, p); for(k = 3, kmax, lucas3 = lucas1 + lucas2; p = 2*k + 1; if(isprime(p) && !(lucas3 % p), print1(p, ", ")); lucas1 = lucas2; lucas2 = lucas3);} \\ _Amiram Eldar_, Jun 06 2024
%Y A122869 Cf. A000032, A000045, A053032, A076518, A122870.
%Y A122869 Subsequence of A002145, A003626, A040105, A040147 and A064739.
%K A122869 nonn
%O A122869 1,1
%A A122869 _Alexander Adamchuk_, Sep 16 2006