A340053 First of three consecutive primes p,q,r such that p*q*r mod (p+q), p*q*r mod (p+r) and p*q*r mod (q+r) are all primes.

13, 17, 37, 41, 43, 73, 89, 103, 127, 151, 157, 167, 239, 257, 271, 281, 337, 353, 379, 401, 409, 419, 421, 443, 521, 557, 587, 601, 607, 683, 719, 743, 839, 883, 941, 953, 983, 1093, 1097, 1297, 1361, 1409, 1427, 1429, 1511, 1543, 1583, 1597, 1637, 1657, 1741, 1801, 1913, 1973, 1993, 2017, 2281

Offset

1,1

Links

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

Example

a(3) = 37 is a term because 37, 41, 43 are consecutive primes, 37*41*43 = 65231, and 65231 mod (37+41) = 23, 65231 mod (37+43) = 31, and 65321 mod (41+43) = 47 are all primes.

Maple

q:= 2: r:= 3:
R:= NULL: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if  isprime(p*q*r mod (p+q)) and isprime(p*q*r mod (p+r)) and isprime(p*q*r mod (q+r)) then
    count:= count+1; R:= R, p;
  fi
od:
R;

Mathematica

tcpQ[{a_, b_, c_}]:=Module[{t=a b c}, AllTrue[{Mod[t, a+b], Mod[t, b+c], Mod[t, a+c]}, PrimeQ]]; Select[Partition[Prime[Range[400]], 3, 1], tcpQ][[;; , 1]] (* Harvey P. Dale, Jul 21 2023 *)

Prog

(PARI) isok(p) = if (isprime(p), my(q=nextprime(p+1), r = nextprime(q+1), pqr = p*q*r); isprime(pqr % (p+q)) && isprime(pqr % (p+r)) && isprime(pqr % (q+r))); \\ Michel Marcus, Dec 28 2020

Crossrefs

Sequence in context: A377465 A069485 A263725 * A174056 A307894 A322472

Adjacent sequences: A340050 A340051 A340052 * A340054 A340055 A340056

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Dec 27 2020

Status

approved