The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 Table of n, a(n) for n=1..15. 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 Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356751. Sequence in context: A085634 A157478 A297583 * A088903 A094205 A136091 Adjacent sequences: A356753 A356754 A356755 * A356757 A356758 A356759 KEYWORD nonn,more AUTHOR Marco Ripà, Aug 26 2022 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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