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.

1885, 4433, 13949, 30709, 39479, 41287, 53627, 55709, 56173, 61957, 63779, 64897, 78217, 79553, 85951, 90097, 92983, 97679, 99517, 101491, 101803, 102131, 103621, 107821, 115915, 119153, 121481, 121619, 128573, 135439, 141349, 141607, 143117, 145337, 146497, 146557, 148219, 152233, 159619, 164083

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.

Maple

N:= 10^6: # for terms <= N
P:= select(isprime, [seq(i, i=3..floor(sqrt(N)), 2)]):
R:= NULL:
for i from 1 to nops(P) do
  p:= P[i];
  for j from 1 to i-1 do
    q:= P[j];
    if 3*q*p > N then break fi;
    for k from 1 to j-1 do
      r:= P[k];
      if r*q*p > N or r >= p*q then break fi;
      s:= p*q mod r; t:= p*r mod q; u:= q*r mod p;
      if isprime(s) and isprime(t) and isprime(u) and isprime(s+t+u)
          then R:= R, p*q*r
        fi;
od od od:
sort([R]);

Mathematica

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 *)

Crossrefs

Subset of A338704.

Sequence in context: A222799 A252111 A259960 * A236613 A190133 A029564

Adjacent sequences: A338702 A338703 A338704 * A338706 A338707 A338708

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Nov 05 2020

Status

approved