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A356751
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Positive integers m such that x^2 - x + m contains more than m/2 prime numbers for x = 1, 2, ..., m.
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1
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3, 5, 7, 11, 17, 41, 47, 59, 67, 101, 107, 161, 221, 227, 347, 377
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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q[k_] := Count[Range[k], _?(PrimeQ[#^2 - # + k] &)] > k/2; Select[Range[400], q] (* Amiram Eldar, Aug 26 2022 *)
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PROG
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(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
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CROSSREFS
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Cf. A014556 (Euler's "Lucky" numbers).
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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