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 A122870 #36 Jun 23 2025 02:59:53
%S A122870 3,7,23,43,47,67,83,103,107,127,163,167,223,227,263,283,307,347,367,
%T A122870 383,443,463,467,487,503,523,547,563,587,607,643,647,683,727,743,787,
%U A122870 823,827,863,883,887,907,947,967,983,1063,1087,1103,1123,1163,1187,1223
%N A122870 Primes congruent to 3 or 7 mod 20.
%C A122870 The old name was "Primes p that divide Lucas((p+1)/2) = A000032((p+1)/2)".
%C A122870 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 (under the old definition above) 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 == 3, 7 (mod 20). - _Jianing Song_, Jun 20 2025
%D A122870 David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.
%H A122870 Vincenzo Librandi, <a href="/A122870/b122870.txt">Table of n, a(n) for n = 1..1000</a>
%H A122870 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%H A122870 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%H A122870 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian Prime</a>.
%t A122870 Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1+1)/2-1]+Fibonacci[(#1+1)/2+1])/#1]&]
%t A122870 Select[Prime[Range[300]], MemberQ[{3, 7}, Mod[#, 20]]&] (* _Vincenzo Librandi_, Jan 06 2013 *)
%o A122870 (Magma) [p: p in PrimesUpTo(1500) | p mod 20 in [3, 7]]; // _Vincenzo Librandi_, Jan 06 2013
%Y A122870 Cf. A000032, A000045, A122869, A216815.
%Y A122870 Subseqeunce of A002145, A003631, A049098, A053027. Essentially the same as A106865.
%K A122870 nonn
%O A122870 1,1
%A A122870 _Alexander Adamchuk_, Sep 16 2006
%E A122870 I merged A216816 into this entry at the suggestion of _Jianing Song_, Jun 20 2025. - _N. J. A. Sloane_, Jun 22 2025