A347531 First of three consecutive primes p,q,r such that r^2-p^2+p, r^2-p^2+q and r^2-p^2+r are consecutive primes.

5, 19, 31, 97, 107, 1429, 1597, 1997, 2441, 2917, 3203, 3581, 3659, 3691, 3847, 7481, 8237, 8377, 8737, 9697, 11467, 12107, 12251, 14071, 17317, 17659, 18481, 18583, 21383, 21491, 23189, 24799, 30931, 33563, 34337, 37507, 38317, 38639, 38707, 39397, 40483, 41221, 43577, 43649, 45121, 46411, 51059

Offset

1,1

Comments

Dickson's conjecture implies that there are infinitely many terms with, for example, the first three consecutive primes p, p+2, p+6 and the second three 13*p+36, 13*p+38 and 13*p+42.

Links

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

Example

a(3) = 31 is a term because 31, 37, 41 are consecutive primes and 41^2-31^2+31 = 751, 41^2-31^2+37 = 757, 41^2-31^2+41 = 761 are consecutive primes.

Maple

R:= NULL: count:= 0:
q:= 2: r:= 3:
while count < 50 do
  p:= q; q:= r; r:= nextprime(r);
  d:= r^2 - p^2;
  if isprime(p+d) and nextprime(p+d)=q+d and nextprime(q+d)=r+d then
    count:= count+1; R:= R, p
  fi
od:
R;

Crossrefs

Sequence in context: A138242 A163076 A122729 * A356716 A262700 A243269

Adjacent sequences: A347528 A347529 A347530 * A347532 A347533 A347534

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Mar 10 2022

Status

approved