%I #18 Jun 03 2024 03:48:10
%S 2,12,65,388,2708,19969,155369,1250182,10345920,87545946,753285178,
%T 6571105993
%N a(n) is the number of primes p less than or equal to 10^n such that p+2 has only prime factors congruent to -1 modulo 4.
%H Gareth A. Jones and Alexander K. Zvonkin, <a href="https://arxiv.org/abs/2401.00270">A number-theoretic problem concerning pseudo-real Riemann surfaces</a>, arXiv:2401.00270 [math.NT], 2023. See Table 5 at page 14.
%e a(2) = 12 since there are 12 primes p less than or equal to 10^2 such that p+2 has only prime factors congruent to -1 modulo 4 (cf. A369105): 5, 7, 17, 19, 29, 31, 41, 47, 61, 67, 79, 97.
%t a[n_] := Length[Select[Prime[Range[10^n]], PrimeQ[f=First/@FactorInteger[#+2]] == Table[True, {j,PrimeNu[#+2]}] && Mod[f,4] == Table[3, {m,PrimeNu[#+2]}] && #<=10^n &]]; Array[a,10]
%o (PARI) is1(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(p[i] % 4 == 1, return(0))); 1;};
%o lista(nmax) = {my(c = 0, pow = 10, n = 1, nm = nmax + 1); forprime(p = 3, , if(p > pow, print1(c, ", "); pow *= 10; n++; if(n == nm, break)); if(is1(p+2), c++));} \\ _Amiram Eldar_, Jun 03 2024
%Y Cf. A001221, A004614, A004767, A027748, A369105, A369107, A369108, A369109.
%K nonn,more
%O 1,1
%A _Stefano Spezia_, Jan 13 2024
%E a(11)-a(12) from _Amiram Eldar_, Jun 03 2024