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A356756 Positive integers m such that x^2 + x + m contains at least m/2 prime numbers for x = 1, 2, ..., m.

%I A356756 #35 Sep 05 2022 22:22:39
%S A356756 1,5,11,17,41,47,59,67,101,107,161,221,227,347,377
%N A356756 Positive integers m such that x^2 + x + m contains at least m/2 prime numbers for x = 1, 2, ..., m.
%C A356756 This sequence is related to A188424, since we are considering only the addends m := 2n - 1 of k^2 + k + 2n - 1 such that A188424(n)/(2n - 1) > 1/2.
%C A356756 It is not a subsequence of A356751, nor vice versa, since 1 is a peculiar term, whereas 3 and 7 do not belong to the present sequence, even if they are terms of A356751.
%C A356756 Furthermore, it is conjectured that the present sequence consists of only 15 terms (it has been checked by brute force that there are only 15 terms which are smaller than 20000). Lastly, we trivially point out that all terms must be odd, since if m is even, then x^2 + x + m is also even.
%C A356756 We trivially note that all the terms are odd (since x^2 + x + 2 is not prime for x = 1, nor for x = 2) and a(n - 1) = A356751(n) holds for every n > 3.
%C A356756 For an explanation of the abundance of primes of the form x^2 + x + m, for some given m, see Goudsmit's paper in Links.
%H A356756 S. A. Goudsmit, <a href="https://doi.org/10.1038/2141164b0">Unusual Prime Number Sequences</a>, Nature Vol. 214 (1967), 1164.
%e A356756 17 is a term since x^2 + x + 17 is prime for x = 1, 2, 3, ..., 15, which is 15 values of x, and 15 >= 17/2.
%t A356756 q[k_] := Count[Range[k], _?(PrimeQ[#^2 + # + k] &)] >= k/2; Select[Range[400], q] (* _Amiram Eldar_, Aug 26 2022 *)
%o A356756 (Python)
%o A356756 from sympy import isprime
%o A356756 def ok(m): return 2*sum(1 for x in range(1, m+1) if isprime(x**2+x+m)) >= m
%o A356756 print([m for m in range(1, 400) if ok(m)]) # _Michael S. Branicky_, Aug 26 2022
%o A356756 (PARI) isok(m) = sum(k=1, m, isprime(k^2 + k + m)) > m/2; \\ _Michel Marcus_, Aug 26 2022
%Y A356756 Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356751.
%K A356756 nonn,more
%O A356756 1,2
%A A356756 _Marco Ripà_, Aug 26 2022