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

1, 5, 11, 17, 41, 47, 59, 67, 101, 107, 161, 221, 227, 347, 377

Offset: 1

Author

Marco Ripà, Aug 26 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.

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.

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.

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.

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.

Examples

			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.
		

Links

• S. A. Goudsmit, Unusual Prime Number Sequences, Nature Vol. 214 (1967), 1164.

Crossrefs

Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356751.

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