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 A338705 Products p*q*r of three distinct primes such that s=(p*q) mod r, t=(p*r) mod q and u=(q*r) mod p, and s+t+u are all prime. 2

%I

%S 1885,4433,13949,30709,39479,41287,53627,55709,56173,61957,63779,

%T 64897,78217,79553,85951,90097,92983,97679,99517,101491,101803,102131,

%U 103621,107821,115915,119153,121481,121619,128573,135439,141349,141607,143117,145337,146497,146557,148219,152233,159619,164083

%N Products p*q*r of three distinct primes such that s=(p*q) mod r, t=(p*r) mod q and u=(q*r) mod p, and s+t+u are all prime.

%H Robert Israel, <a href="/A338705/b338705.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3)=13949 is a member because 13949 = 13*29*37 with s = (13*29) mod 37 = 7, t = (13*37) mod 29 = 17, u = (29*37) mod 13 = 7, and 7+17+7 = 31, all prime.

%p N:= 10^6: # for terms <= N

%p P:= select(isprime, [seq(i,i=3..floor(sqrt(N)),2)]):

%p R:= NULL:

%p for i from 1 to nops(P) do

%p p:= P[i];

%p for j from 1 to i-1 do

%p q:= P[j];

%p if 3*q*p > N then break fi;

%p for k from 1 to j-1 do

%p r:= P[k];

%p if r*q*p > N or r >= p*q then break fi;

%p s:= p*q mod r; t:= p*r mod q; u:= q*r mod p;

%p if isprime(s) and isprime(t) and isprime(u) and isprime(s+t+u)

%p then R:= R, p*q*r

%p fi;

%p od od od:

%p sort([R]);

%t Block[{a = {}, nn = 164500}, Do[Do[Do[If[And[Length@ Union[{#1, #2, #3}] == 3, AllTrue[{##}~Join~{#1 + #2 + #3} & @@ {Mod[#1 #2, #3], Mod[#1 #3, #2], Mod[#2 #3, #1]}, PrimeQ]], AppendTo[a, #1 #2 #3]] & @@ {Prime[i], Prime[j], Prime[k]}, {k, j - 1}], {j, i - 1}], {i, PrimePi@ Floor[Sqrt[nn]]}]; TakeWhile[Union@ a, # <= nn &]] (* _Michael De Vlieger_, Nov 05 2020 *)

%Y Subset of A338704.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Nov 05 2020

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Last modified December 6 13:05 EST 2021. Contains 349563 sequences. (Running on oeis4.)