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

1,1

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, Entry points and periods of Lucas sequences

Eric Weisstein's World of Mathematics, Lucas Number.

Eric Weisstein's World of Mathematics, Gaussian Prime.

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 *)

Prog

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

Crossrefs

Cf. A000032, A000045, A122869, A216815.

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

Sequence in context: A029932 A084739 A133434 * A216816 A079477 A014426

Adjacent sequences: A122867 A122868 A122869 * A122871 A122872 A122873

Keyword

nonn

Author

Alexander Adamchuk, Sep 16 2006

Extensions

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

Status

approved