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

Author

David James Sycamore, Nov 11 2018

Keywords

• nonn

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).

Examples

			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.
		

Crossrefs

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

Programs

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[qAmiram Eldar, Nov 12 2018 *)
Join[{5}, Select[Prime[Range[400]], PrimeQ[((2 # + 1)) / 3] &]] (* Vincenzo Librandi, Nov 17 2018 *)

PARI

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

Formula

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