A352803 a(n) is the first prime p such that, with q the next prime, p^2+q is 10^n times a prime.

2, 523, 2243, 39419, 763031, 37427413, 594527413, 5440486343, 1619625353, 35960850223, 17012632873031, 43502632873031, 2322601810486343, 5470654702304929, 99466287423954043, 1917321601810486343, 6091565756519625353

Offset: 0

Author

J. M. Bergot and Robert Israel, Apr 05 2022

Keywords

• nonn
• more

Comments

From Daniel Suteu, Dec 28 2022: (Start)

For n >= 1, a(n) has the form k * 10^n + x, for some k >= 0, where x is a solution to the modular quadratic equation x^2 + x + d == 0 (mod 10^n), where d = q-p.

a(17) <= 379430283012423635659, a(18) <= 1857717470295105527413. (End)

Examples

			a(2) = 2243 because 2243 is prime, the next prime is 2251, 2243^2+2251 = 5033300 = 10^2*50333 and 50333 is prime.
		

Crossrefs

Cf. A002386, A352848, A352852.

Programs

Maple

V:= Array(0..5):
count:= 0:
q:= 2:
while count < 6 do
  p:= q; q:= nextprime(p);
  v:= p^2+q;
  r:= padic:-ordp(v, 2);
  if r <= 5 and V[r] = 0 and padic:-ordp(v, 5) = r and isprime(v/10^r) then
     V[r]:= p; count:= count+1;
  fi;
od:
convert(V, list);

PARI

isok(n,p,q) = my(v=valuation(p^2+q, 10)); (v == n) && isprime((p^2+q)/10^v);
a(n) = my(p=2); forprime(q=p+1, oo, if(isok(n,p,q), return(p)); p=q); \\ Daniel Suteu, Apr 07 2022

Extensions

a(6)-a(9) from Daniel Suteu, Apr 07 2022

a(10)-a(16) from Daniel Suteu, Dec 28 2022