A122870 Primes congruent to 3 or 7 mod 20.

3, 7, 23, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 263, 283, 307, 347, 367, 383, 443, 463, 467, 487, 503, 523, 547, 563, 587, 607, 643, 647, 683, 727, 743, 787, 823, 827, 863, 883, 887, 907, 947, 967, 983, 1063, 1087, 1103, 1123, 1163, 1187, 1223

Offset: 1

Author

Alexander Adamchuk, Sep 16 2006

Keywords

• nonn

Comments

The old name was "Primes p that divide Lucas((p+1)/2) = A000032((p+1)/2)".

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

References

• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989; see p. 33.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jianing Song, Lucas sequences and entry point modulo p
• Eric Weisstein's World of Mathematics, Lucas Number.
• Eric Weisstein's World of Mathematics, Gaussian Prime.

Crossrefs

Cf. A000032, A000045, A122869, A216815.

Subseqeunce of A002145, A003631, A049098, A053027. Essentially the same as A106865.

Programs

Magma

[p: p in PrimesUpTo(1500) | p mod 20 in [3, 7]]; // Vincenzo Librandi, Jan 06 2013

Mathematica

Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1+1)/2-1]+Fibonacci[(#1+1)/2+1])/#1]&]
Select[Prime[Range[300]], MemberQ[{3, 7}, Mod[#, 20]]&] (* Vincenzo Librandi, Jan 06 2013 *)

Extensions

I merged A216816 into this entry at the suggestion of Jianing Song, Jun 20 2025. - N. J. A. Sloane, Jun 22 2025