A007635 -id:A007635 - cp's OEIS frontend

A318791 Prime generating polynomial: a(n) = 9n^2 - 249n + 1763.

1523, 1301, 1097, 911, 743, 593, 461, 347, 251, 173, 113, 71, 47, 41, 53, 83, 131, 197, 281, 383, 503, 641, 797, 971, 1163, 1373, 1601, 1847, 2111, 2393, 2693, 3011, 3347, 3701, 4073, 4463, 4871, 5297, 5741, 6203, 6683, 7181, 7697, 8231, 8783, 9353

Offset: 1

Arashdeep Singh, Dec 15 2018

This polynomial (9*n^2 - 249*n + 1763) generates 40 distinct primes in succession from n = 1 to 40.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005846, A007635, A048059, A202018.

seq(9*n^2-249*n+1763,n=1..50); # Muniru A Asiru, Dec 19 2018

Array[9#^2 - 249# + 1763 &, 50] (* Amiram Eldar, Dec 15 2018 *)

From Chai Wah Wu, Feb 12 2019: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f.: x*(-1763*x^2 + 3268*x - 1523)/(x - 1)^3. (End)
a(n) = p(41 - 3*n), where p(n) = n^2 + n + 41 is Euler's prime generating polynomial - see A202018 and A005846. - Peter Bala, Jun 10 2021
E.g.f.: exp(x)*(9*x^2 - 240*x + 1763) - 1763. - Elmo R. Oliveira, Feb 10 2025

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

A173753 Let f(j) = j^2 + j + 17 and g(j) = j^2 - j + 17. Sequence gives pi(f(j)) - pi(g(j)) as j runs through those nonnegative integers for which both f(j) and g(j) are prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 4, 3, 4, 3, 4, 5, 6, 3, 6, 6, 7, 5, 9, 9, 8, 9, 7, 8, 15, 13, 12, 11, 14, 16, 14, 16, 17, 20, 19, 23, 17, 20, 26, 22, 27, 30, 23, 25, 28, 26, 32, 36, 26, 35, 25, 30, 31, 33, 34, 33, 40, 41, 36, 39, 43, 36, 40, 41, 49, 43, 48, 47, 51, 55, 53, 47, 58, 54, 56, 63, 60

Offset: 1

Juri-Stepan Gerasimov, Feb 23 2010

			a(1) = 7 - 7 = 0 where 0^2 + 0 + 17 = 17 = prime(7) and 0^2 - 0 + 17 = 17 = prime(7);
a(2) = 8 - 7 = 1 where 1^2 + 1 + 17 = 19 = prime(8) and 1^2 - 1 + 17 = 17 = prime(7);
a(3) = 9 - 8 = 1 where 2^2 + 2 + 17 = 23 = prime(9) and 2^2 - 2 + 17 = 19 = prime(8);
a(4) = 10 - 9 = 1 where 3^2 + 3 + 17 = 29 = prime(10) and 3^2 - 3 + 17 = 23 = prime(9).

Cf. A007635.

for x from 0 to 1000 do mp := x^2+x+17 ; kp := x^2-x+17 ; if isprime(mp) and isprime(kp) then m := numtheory[pi](mp) ; k := numtheory[pi](kp) ; printf("%d,",m-k) ; end if; end do : # R. J. Mathar, Mar 01 2010

f[n_]:=Module[{c=n^2+17,a,b},a=c+n;b=c-n;If[And@@PrimeQ[{a,b}], PrimePi[a]- PrimePi[b],0]]; Join[{0},Select[Array[f,400,0],#!=0&]] (* Harvey P. Dale, Jul 13 2011 *)

a(31) and a(33) corrected and sequence extended by R. J. Mathar, Mar 01 2010

Name edited by Jon E. Schoenfield, Jan 30 2019

A289839 Primes of the form 8n^2+8n+31.

Original entry on oeis.org

31, 47, 79, 127, 191, 271, 367, 479, 607, 751, 911, 1087, 1279, 1487, 1951, 2207, 2767, 3391, 3727, 4079, 4447, 4831, 5231, 5647, 6079, 6991, 9007, 9551, 10111, 10687, 11279, 11887, 12511, 13151, 13807, 14479, 17327, 20431, 21247, 22079, 24671, 26479, 27407

Offset: 1

Waldemar Puszkarz, Oct 06 2017

nonn

The first 14 terms correspond to n from 0 to 13, which makes 8*n^2+8*n+31 a prime-generating polynomial (see the link).
This is a prime-generating polynomial of the form c*n^2+c*n+p, where c=2^k (k=0,1,2...) and p is prime with c and p containing at most two digits. Prime-generating polynomials of this kind arise for k=0,1,2,3 - see A005846 and A007635 (k=0), A007639 (k=1), and A048988 (k=2).
All terms are of the form 4m+3. Terms 1 and 4 are Mersenne primes (A000668).

			79 is a term as it is a prime corresponding to n=2: 8*4+8*2+31=79.

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A000040 (primes), A005846, A007635, A007639, A048988, A281437, A292578 (similar prime-generating sequences).

Select[Range[0,100]//8#^2+8#+31&, PrimeQ]

for(n=0, 100, isprime(p=8*n^2+8*n+31)&& print1(p ", "))

cp's OEIS Frontend

A318791 Prime generating polynomial: a(n) = 9*n^2 - 249*n + 1763.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A173753 Let f(j) = j^2 + j + 17 and g(j) = j^2 - j + 17. Sequence gives pi(f(j)) - pi(g(j)) as j runs through those nonnegative integers for which both f(j) and g(j) are prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A289839 Primes of the form 8*n^2+8*n+31.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A318791 Prime generating polynomial: a(n) = 9n^2 - 249n + 1763.

A289839 Primes of the form 8n^2+8n+31.