A352852 -id:A352852 - cp's OEIS frontend

A352126 Primes p such that, if q is the next prime, both p+q^2 and p^2+q are primes times powers of 10.

2, 4806589, 8369989, 11168569, 20666869, 25068349, 25465249, 29046469, 37597849, 40593349, 44242669, 45405889, 47975869, 49637149, 50057569, 51468349, 57060469, 59570449, 64602589, 64707889, 65940769, 70752049, 75879169, 81799789, 87845869, 90277249, 92415649, 93315889, 95458249, 97225069

Offset: 1

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

nonn

Primes prime(k) such that when any trailing zeros are removed from A349660(k) and A352851(k), the results are prime.

Except for 2, each term and the next prime == 19 (mod 30).

			a(3) = 8369989 is a term because it is prime, the next prime is 8370049,
8369989+8370049^2 = 70057728632390, 8369989^2+8370049 = 70056724230170, and 7005772863239 and 7005672423017 are prime.

Robert Israel, Table of n, a(n) for n = 1..250

Cf. A349660, A352851.

Intersection of A352837 and A352852.

R:= NULL: count:= 0:
q:= 2:
while count < 30 do
  p:= q; q:= nextprime(p);
  w:= p+q^2;
  m:= padic:-ordp(w,2);
  if padic:-ordp(w,5) <> m then next fi;
  if m > 0 then w:= w/10^m fi;
  if not isprime(w) then next fi;
  v:= p^2+q;
  m:= padic:-ordp(v,2);
  if padic:-ordp(v,5) <> m then next fi;
  if m > 0 then v:= v/10^m fi;
  if isprime(v) then count:= count+1; R:= R, p; fi
od:
R;

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

Original entry on oeis.org

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

Offset: 0

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

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)

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

Cf. A002386, A352848, A352852.

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);

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

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

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

cp's OEIS Frontend

A352126 Primes p such that, if q is the next prime, both p+q^2 and p^2+q are primes times powers of 10.

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