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

3, 5, 7, 11, 17, 41, 47, 59, 67, 101, 107, 161, 221, 227, 347, 377

Offset: 1

Author

Marco Ripà, Aug 25 2022

Keywords

• nonn
• more

Comments

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.

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).

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.

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

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

Examples

			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.
		

Links

• S. A. Goudsmit, Unusual Prime Number Sequences, Nature Vol. 214 (1967), 1164.
• Brady Haran and Matt Parker, Caboose Numbers, Youtube video, June 2024.

Crossrefs

Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356756.

Cf. A014556 (Euler's "Lucky" numbers).

Programs

Mathematica

q[k_] := Count[Range[k], ?(PrimeQ[#^2 - # + k] &)] > k/2; Select[Range[400], q] (* _Amiram Eldar, Aug 26 2022 *)

PARI

isok(m) = sum(k=1, m, isprime(k^2 - k + m)) > m/2; \\ Michel Marcus, Aug 26 2022

Python

from sympy import isprime
def ok(m): return 2*sum(1 for x in range(1, m+1) if isprime(x**2-x+m)) > m
print([m for m in range(1, 400) if ok(m)]) # Michael S. Branicky, Aug 26 2022