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A356751 Positive integers m such that x^2 - x + m contains more than m/2 prime numbers for x = 1, 2, ..., m.

%I A356751 #63 Jun 20 2024 16:26:36
%S A356751 3,5,7,11,17,41,47,59,67,101,107,161,221,227,347,377
%N A356751 Positive integers m such that x^2 - x + m contains more than m/2 prime numbers for x = 1, 2, ..., m.
%C A356751 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 A356751 Furthermore, it is conjectured that the present sequence consists of only 16 terms (it has been checked by brute force that there are only 16 terms which are smaller than 20000) and that they are all prime or semiprime (e.g., a(12) = 161, a(13) = 221, and a(16) = 377 are semiprime). Lastly, we trivially point out that all terms must be odd, since if m is even, then x^2 - x + m is also even (and x^2 - x + 2 has only one prime for x <= 2).
%C A356751 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.
%C A356751 Stronger conjecture: for every real number e > 0 and every integer m > 0, there are finitely many integer polynomials P(x) = Ax^2 + Bx + C with at least e*m primes in P(1), ..., P(m) and max(|A|, |B|, |C|) <= m. - _Charles R Greathouse IV_, Sep 11 2022
%C A356751 Altering the bounds for x in the definition to 0 <= x <= m-1 (and counting the same prime twice for x=0 and x=1 if m is prime) would result in an additional term 2. Conjecturally, there would be no more additional terms. - _Pontus von Brömssen_, Jun 20 2024
%H A356751 S. A. Goudsmit, <a href="https://doi.org/10.1038/2141164b0">Unusual Prime Number Sequences</a>, Nature Vol. 214 (1967), 1164.
%H A356751 Brady Haran and Matt Parker, <a href="https://www.youtube.com/watch?v=gM5uNcgn2NQ">Caboose Numbers</a>, Youtube video, June 2024.
%e A356751 7 is a term since x^2 - x + 7 is prime for x = 1, 3, 4, and 6, which is 4 values of x, and 4 > 7/2.
%t A356751 q[k_] := Count[Range[k], _?(PrimeQ[#^2 - # + k] &)] > k/2; Select[Range[400], q] (* _Amiram Eldar_, Aug 26 2022 *)
%o A356751 (PARI) isok(m) = sum(k=1, m, isprime(k^2 - k + m)) > m/2; \\ _Michel Marcus_, Aug 26 2022
%o A356751 (Python)
%o A356751 from sympy import isprime
%o A356751 def ok(m): return 2*sum(1 for x in range(1, m+1) if isprime(x**2-x+m)) > m
%o A356751 print([m for m in range(1, 400) if ok(m)]) # _Michael S. Branicky_, Aug 26 2022
%Y A356751 Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356756.
%Y A356751 Cf. A014556 (Euler's "Lucky" numbers).
%K A356751 nonn,more
%O A356751 1,1
%A A356751 _Marco Ripà_, Aug 25 2022