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

%I #23 Dec 24 2018 08:28:51

%S 5,7,19,43,61,79,109,151,163,223,271,349,421,439,523,601,613,631,673,

%T 691,811,853,919,991,1009,1051,1063,1153,1213,1231,1279,1321,1429,

%U 1531,1549,1663,1693,1789,1801,1873,1933,1951,2113,2143,2179,2221,2239,2503,2539,2683,2791,2833,2851

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

%C 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).

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

%e 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.

%e For p = 5, q = 2 (prime); 2*3 = 6 == 1 (mod 5) therefore 5 is a term.

%p for n from 3 to 300 do

%p Y := ithprime(n);

%p Z := 1/3 mod Y;

%p if isprime(Z) then print(Y);

%p end if:

%p end do:

%t 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 *)

%t Join[{5}, Select[Prime[Range[400]], PrimeQ[((2 # + 1)) / 3] &]] (* _Vincenzo Librandi_, Nov 17 2018 *)

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

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

%K nonn

%O 1,1

%A _David James Sycamore_, Nov 11 2018

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Last modified April 25 03:15 EDT 2024. Contains 371964 sequences. (Running on oeis4.)