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%I #37 Dec 29 2022 09:16:48
%S 2,523,2243,39419,763031,37427413,594527413,5440486343,1619625353,
%T 35960850223,17012632873031,43502632873031,2322601810486343,
%U 5470654702304929,99466287423954043,1917321601810486343,6091565756519625353
%N a(n) is the first prime p such that, with q the next prime, p^2+q is 10^n times a prime.
%C From _Daniel Suteu_, Dec 28 2022: (Start)
%C 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.
%C a(17) <= 379430283012423635659, a(18) <= 1857717470295105527413. (End)
%e a(2) = 2243 because 2243 is prime, the next prime is 2251, 2243^2+2251 = 5033300 = 10^2*50333 and 50333 is prime.
%p V:= Array(0..5):
%p count:= 0:
%p q:= 2:
%p while count < 6 do
%p p:= q; q:= nextprime(p);
%p v:= p^2+q;
%p r:= padic:-ordp(v, 2);
%p if r <= 5 and V[r] = 0 and padic:-ordp(v, 5) = r and isprime(v/10^r) then
%p V[r]:= p; count:= count+1;
%p fi;
%p od:
%p convert(V, list);
%o (PARI)
%o isok(n,p,q) = my(v=valuation(p^2+q, 10)); (v == n) && isprime((p^2+q)/10^v);
%o a(n) = my(p=2); forprime(q=p+1, oo, if(isok(n,p,q), return(p)); p=q); \\ _Daniel Suteu_, Apr 07 2022
%Y Cf. A002386, A352848, A352852.
%K nonn,more
%O 0,1
%A _J. M. Bergot_ and _Robert Israel_, Apr 05 2022
%E a(6)-a(9) from _Daniel Suteu_, Apr 07 2022
%E a(10)-a(16) from _Daniel Suteu_, Dec 28 2022
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