A338704 Products p*q*r of three distinct primes such that (p*q) mod r, (p*r) mod q and (q*r) mod p are all prime.

1023, 1885, 2635, 3857, 4433, 4623, 5883, 7579, 7611, 8987, 9447, 11607, 13949, 14053, 14573, 14839, 14965, 15189, 15265, 16287, 17507, 19599, 20661, 21535, 22119, 23433, 24827, 24963, 25359, 25517, 26781, 30385, 30709, 31537, 34715, 36499, 38121, 38315, 38533, 39479, 39867, 41287, 41915, 42107

Offset

1,1

Links

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

Example

a(3)=2635 is a term because 2635=5*17*31 with (5*17) mod 31 = 23, (5*31) mod 17 = 2 and (17*31) mod 5 = 2 all prime.

Maple

N:= 10^5: # 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;
      if isprime(p*q mod r) and isprime(p*r mod q) and isprime(q*r mod p) then
         R:=R, p*q*r;
      fi
od od od:
sort([R]);

Mathematica

Block[{a = {}, nn = 42500}, Do[Do[Do[If[And[Length@ Union[{#1, #2, #3}] == 3, AllTrue[{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 *)
Select[Union[Times@@@Select[Subsets[Prime[Range[50]], {3}], AllTrue[{ Mod[ #[[1]]#[[2]], #[[3]]], Mod[#[[2]]#[[3]], #[[1]]], Mod[#[[1]]#[[3]], #[[2]]]}, PrimeQ]&]], #<=50000&] (* Harvey P. Dale, Aug 11 2021 *)

Crossrefs

Contains A338705.

Sequence in context: A031969 A166512 A038461 * A158421 A023060 A223079

Adjacent sequences: A338701 A338702 A338703 * A338705 A338706 A338707

Keyword

nonn

Author

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

Status

approved