A321510 Primes p for which there exists a prime q < p such that 3*q == 1 (mod p).

5, 7, 19, 43, 61, 79, 109, 151, 163, 223, 271, 349, 421, 439, 523, 601, 613, 631, 673, 691, 811, 853, 919, 991, 1009, 1051, 1063, 1153, 1213, 1231, 1279, 1321, 1429, 1531, 1549, 1663, 1693, 1789, 1801, 1873, 1933, 1951, 2113, 2143, 2179, 2221, 2239, 2503, 2539, 2683, 2791, 2833, 2851

Offset

1,1

Comments

A104163 with 5 prepended (see example). For any prime p in A104163 q = (2*p+1)/3, then q < p and 3*q == 1 (mod p).

Links

Table of n, a(n) for n=1..53.

Formula

a(n+1) = A104163(n); n >= 1.

Example

For p = 11, the only number t < 11 such that 3*t == 1 (mod 11) is t = 4, which is not prime, therefore 11 is not a term.
For p = 5, q = 2 (prime); 2*3 = 6 == 1 (mod 5) therefore 5 is a term.

Maple

for n from 3 to 300 do
Y := ithprime(n);
Z := 1/3 mod Y;
if isprime(Z) then print(Y);
end if:
end do:

Mathematica

aQ[p_]:=Module[{ans=False, q=2}, While[q<p, If[Mod[3*q, p]==1, ans=True; Break[]]; q=NextPrime[q]]; ans]; Select[Prime[Range[350]], aQ] (* Amiram Eldar, Nov 12 2018 *)
Join[{5}, Select[Prime[Range[400]], PrimeQ[((2 # + 1)) / 3] &]] (* Vincenzo Librandi, Nov 17 2018 *)

Prog

(PARI) isok(p) = if (isprime(p), forprime(q=1, p-1, if ((3*q % p) == 1, return (1)))); \\ Michel Marcus, Nov 14 2018

Crossrefs

Cf. A104163 (essentially the same sequence), A005383.

Sequence in context: A296935 A106987 A342738 * A084198 A036061 A288608

Adjacent sequences: A321507 A321508 A321509 * A321511 A321512 A321513

Keyword

nonn

Author

David James Sycamore, Nov 11 2018

Status

approved