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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. 1
1, 5, 11, 17, 41, 47, 59, 67, 101, 107, 161, 221, 227, 347, 377 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
LINKS
S. A. Goudsmit, Unusual Prime Number Sequences, Nature Vol. 214 (1967), 1164.
EXAMPLE
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.
MATHEMATICA
q[k_] := Count[Range[k], _?(PrimeQ[#^2 + # + k] &)] >= k/2; Select[Range[400], q] (* Amiram Eldar, Aug 26 2022 *)
PROG
(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
(PARI) isok(m) = sum(k=1, m, isprime(k^2 + k + m)) > m/2; \\ Michel Marcus, Aug 26 2022
CROSSREFS
Sequence in context: A085634 A157478 A297583 * A088903 A094205 A136091
KEYWORD
nonn,more
AUTHOR
Marco Ripà, Aug 26 2022
STATUS
approved

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Last modified April 23 13:51 EDT 2024. Contains 371914 sequences. (Running on oeis4.)