A188424 -id:A188424 - cp's OEIS frontend

A302826 a(n) is number of primes of form k^2 + n - k for 0 < k < n.

1, 2, 0, 4, 0, 4, 0, 2, 0, 10, 0, 4, 0, 3, 0, 16, 0, 6, 0, 5, 0, 10, 0, 10, 0, 5, 0, 13, 0, 14, 0, 3, 0, 10, 0, 16, 0, 7, 0, 40, 0, 8, 0, 6, 0, 26, 0, 12, 0, 9, 0, 19, 0, 14, 0, 9, 0, 34, 0, 21, 0, 5, 0, 19, 0, 36, 0, 13, 0, 28, 0, 18, 0, 7, 0, 31, 0, 18, 0, 19, 0, 34, 0, 15

Offset: 2

Seiichi Manyama, Apr 14 2018

nonn

			Primes of form k^2 + n - k for 0 < k < n:
  n\k|  1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16  |a(n)
  ---+------------------------------------------------------------------------+----
    2|  2;                                                                    |  1
    3|  3,  5;                                                                |  2
    4|                                                                        |  0
    5|  5,  7, 11, 17;                                                        |  4
    6|                                                                        |  0
    7|  7,   , 13, 19,   , 37;                                                |  4
    8|                                                                        |  0
    9|   , 11,   ,   , 29,   ,   ,   ;                                        |  2
   10|                                                                        |  0
   11| 11, 13, 17, 23, 31, 41, 53, 67, 83, 101;                               | 10
   12|                                                                        |  0
   13| 13,   , 19,   ,   , 43,   ,   ,   , 103,    ,    ;                     |  4
   14|                                                                        |  0
   15|   , 17,   ,   ,   ,   ,   , 71,   ,    ,    ,    ,    , 197;           |  3
   16|                                                                        |  0
   17| 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257; | 16

Seiichi Manyama, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Lucky Number of Euler
Wikipedia, Lucky numbers of Euler

Cf. A014556, A188424, A302445.

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

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Aug 25 2022

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

			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.

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

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Aug 26 2022

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.

			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.

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

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

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

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

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

cp's OEIS Frontend

A302826 a(n) is number of primes of form k^2 + n - k for 0 < k < n.

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A356751 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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A356756 Positive integers m such that x^2 + x + m contains at least m/2 prime numbers for x = 1, 2, ..., m.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Python