A164926 -id:A164926 - cp's OEIS frontend

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

11, 17, 41, 844427, 51448361, 86966771, 122983031, 180078317, 960959381, 1278189947, 1761702947, 1829187287, 2426256797, 2911675511, 3013107257, 4778888351, 5221343711

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..2713

Generates primes for x=1..k: A001359 (1), A022004 (2), A172454 (3), A187057 (4), A187058 (5), A144051 (6), A187060 (7), A190800 (8), this sequence (9), A191457 (10), A191458 (11), A253592 (12), A253605 (13). Each is by definition a subsequence of preceding sequences.
Subsequence such that x=10 gives a composite number: A211238.
Cf. A160548, A164926.

is(n)=for(x=0,9, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A210360 Prime numbers p such that x^2 + x + p produces primes for x = 0..1 but not x = 2.

Original entry on oeis.org

3, 29, 59, 71, 137, 149, 179, 197, 239, 269, 281, 419, 431, 521, 569, 599, 617, 659, 809, 827, 1019, 1031, 1049, 1061, 1151, 1229, 1289, 1319, 1451, 1619, 1667, 1697, 1721, 1787, 1877, 1931, 1949, 2027, 2087, 2111, 2129, 2141, 2309, 2339, 2381, 2549, 2591

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(2).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210361, A210362, A210363, A210364, A210365.

lookfor = 2; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A210361 Prime numbers p such that x^2 + x + p produces primes for x = 0..2 but not x = 3.

Original entry on oeis.org

107, 191, 311, 461, 821, 857, 881, 1301, 1871, 1997, 2081, 2237, 2267, 2657, 3251, 3461, 3671, 4517, 4967, 5231, 5477, 5501, 5651, 6197, 6827, 7877, 8087, 8291, 8537, 8861, 9431, 10427, 10457, 11171, 12917, 13001, 13691, 13757, 13877, 14081, 14321, 15641

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(3).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210362, A210363, A210364, A210365.

lookfor = 3; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A210362 Prime numbers p such that x^2 + x + p produces primes for x = 0..3 but not x = 4.

Original entry on oeis.org

5, 101, 227, 1091, 1481, 1487, 3917, 4127, 4787, 8231, 9461, 10331, 11777, 12107, 14627, 16061, 20747, 25577, 27737, 29021, 32297, 33347, 35531, 35591, 36467, 38447, 39227, 41177, 42461, 44267, 44531, 49031, 59441, 69191, 77237, 79811, 80777, 93251, 93491

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(4).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210363, A210364, A210365.

lookfor = 4; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[10000]],AllTrue[#+{2,6,12},PrimeQ]&&!PrimeQ[#+20]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 26 2015 *)
Select[Prime[Range[10000]],Boole[PrimeQ[#+{2,6,12,20}]]=={1,1,1,0}&] (* Harvey P. Dale, Nov 17 2024 *)

A210363 Prime numbers p such that x^2 + x + p produces primes for x = 0..4 but not x = 5.

Original entry on oeis.org

347, 641, 1427, 2687, 4001, 4637, 4931, 19421, 21011, 22271, 23741, 26711, 27941, 32057, 43781, 45821, 55331, 55817, 68207, 71327, 91571, 128657, 165701, 167621, 172421, 179897, 191447, 193871, 205421, 234191, 239231, 258107, 258611, 259157, 278807, 290021

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(5).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210362, A210364, A210365.

lookfor = 5; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[26000]],AllTrue[#+{2,6,12,20},PrimeQ] && !PrimeQ[ #+30]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2017 *)
Select[Prime[Range[26000]],Boole[PrimeQ[#+{2,6,12,20,30}]]=={1,1,1,1,0}&] (* Harvey P. Dale, May 29 2025 *)

A210364 Prime numbers p such that x^2 + x + p produces primes for x = 0..5 but not x = 6.

Original entry on oeis.org

1607, 3527, 13901, 31247, 33617, 55661, 68897, 97367, 166841, 195731, 221717, 347981, 348431, 354371, 416387, 506327, 548831, 566537, 929057, 954257, 1246367, 1265081, 1358801, 1505087, 1538081, 1595051, 1634441, 1749257, 2200811, 2322107, 2641547, 2697971

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(6).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210362, A210363, A210365.

lookfor = 6; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A210365 Prime numbers p such that x^2 + x + p produces primes for x = 0..6 but not x = 7.

Original entry on oeis.org

1277, 28277, 113147, 421697, 665111, 1164587, 1615631, 2798921, 2846771, 3053747, 5071667, 5093507, 5344247, 5706641, 6383051, 8396777, 10732817, 10812407, 11920367, 13176587, 16197947, 16462541, 16655447, 16943471, 17807831, 18102101, 20488901, 23421311

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(7).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210362, A210363, A210364.

lookfor = 7; t = {}; n = 0; While[Length[t] < 30, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A370387 a(n) is the least prime p such that p + 6k(k+1) is prime for 0 <= k <= n-1 but not for k=n.

Original entry on oeis.org

2, 19, 5, 67, 7, 281, 1051, 6791, 11, 115599457, 365705201, 79352440891, 286351937491, 5810592517241, 17, 1942721697854617

Offset: 1

J.W.L. (Jan) Eerland, Mar 12 2024

a(10), ..., a(14) > 10^7, a(15) = 17, a(16), ..., a(20) > 10^7.

a(29) = 31. - Chai Wah Wu, Apr 10 2024

Cf. A164926, A371024.

f:= proc(p) local k;
  for k from 1 while isprime(p+k*(k+1)*6) do od:
  k
end proc:
A:= Vector(12): count:= 0:
for i from 1 while count < 12 do
  v:= f(ithprime(i));
  if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi
od:
convert(A,list);

Table[p=1;m=6;Monitor[Parallelize[While[True,If[And[MemberQ[PrimeQ[Table[p+m*k*(k+1),{k,0,n-1}]],False]==False,PrimeQ[p+m*n*(n+1)]==False],Break[]];p++];p],p],{n,1,10}]

isok(p, n) = for (k=0, n-1, if (! isprime(p + 6*k*(k+1)), return(0))); return (!isprime(p + 6*n*(n+1)));
a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p;

use ntheory qw(:all); sub a { my $n = $[0]; my $lo = 2; my $hi = 2*$lo; while (1) { my @terms = grep { !is_prime($ + 6*$n*($n+1)) } sieve_prime_cluster($lo, $hi, map { 6*$*($+1) } 1 .. $n-1); return $terms[0] if @terms; $lo = $hi+1; $hi = 2*$lo; } }; $| = 1; for my $n (1..100) { print a($n), ", " } # Daniel Suteu, Dec 30 2024

a(10)-a(11) from Chai Wah Wu, Apr 10 2024
a(12) from Chai Wah Wu, Apr 11 2024
a(13)-a(14) from David A. Corneth, Apr 11 2024
a(15) from J.W.L. (Jan) Eerland, Mar 12 2024
a(16) from Daniel Suteu, Dec 30 2024

A371024 a(n) is the least prime p such that p + 4k(k+1) is prime for 0 <= k <= n-1 but not for k=n.

Original entry on oeis.org

2, 3, 29, 5, 23, 269, 272879, 149, 61463, 929, 7426253, 2609, 233, 59, 78977932125503

Offset: 1

J.W.L. (Jan) Eerland, Mar 08 2024

a(15) > 3277860277, a(16) > 3103623446, a(17) > 2853255995,
a(18) = 653, a(19) > 2480173428, a(20) > 2058783580, a(21) > 1894529774, a(22) > 1896261075, a(23) > 1836831342, a(24), ..., a(100) > 15000000.
Other than a(1)-a(14) and a(18), no terms < 24870000007. - Michael S. Branicky, Apr 12 2024
From David A. Corneth, Apr 12 2024: (Start)
Using remainders mod q we can restrict the search. For example for a(15) a term can only be 2, 3 or 5 (mod 7). Or maybe 7 itself. If a(15) = p == 1 (mod 7) then for k = 3 we have q + 4*3*(3+1) == 0 mod 7. Similarily number 0, 4 and 6 (mod 7) produce a multiple of 7 where they should not.
Doing so for various primes mod q we can reduce the number of remainders and with that the search space by combining the possible remainders using the Chinese Remainder Theorem (CRT).
So the possible remainders mod 2 are 1. The possible remainders mod 3 are 2. Using the CRT, a number of the form 1 (mod 2) and 2 (mod 3) simultaneously is of the form 5 (mod 6).
a(15) > 2.3*10^13 if it exists. (End)

Cf. A164926, A370387.

f:= proc(p) local k;
  for k from 1 while isprime(p+k*(k+1)*4) do od:
  k
end proc:
A:= Vector(12): count:= 0:
for i from 1 while count < 12 do
  v:= f(ithprime(i));
  if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi
od:
convert(A,list);

Table[p=1;m=4;Monitor[Parallelize[While[True,If[And[MemberQ[PrimeQ[Table[p+m*k*(k+1),{k,0,n-1}]],False]==False,PrimeQ[p+m*n*(n+1)]==False],Break[]];p++];p],p],{n,1,10}]

isok(p, n) = for (k=0, n-1, if (! isprime(p + 4*k*(k+1)), return(0))); return (!isprime(p + 4*n*(n+1)));
a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Mar 12 2024

use ntheory qw(:all); sub a { my $n = $[0]; my $lo = 2; my $hi = 2*$lo; while (1) { my @terms = grep { !is_prime($ + 4*$n*($n+1)) } sieve_prime_cluster($lo, $hi, map { 4*$*($+1) } 1 .. $n-1); return $terms[0] if @terms; $lo = $hi+1; $hi = 2*$lo; } }; $| = 1; for my $n (1..100) { print a($n), ", " }; # Daniel Suteu, Dec 17 2024

from sympy import isprime, nextprime
from itertools import count, islice
def f(p):
    k = 1
    while isprime(p+4*k*(k+1)): k += 1
    return k
def agen(verbose=False): # generator of terms
    adict, n, p = dict(), 1, 1
    while True:
        p = nextprime(p)
        v = f(p)
        if v not in adict:
            adict[v] = p
            if verbose: print("FOUND", v, p)
        while n in adict:
            yield adict[n]; n += 1
print(list(islice(agen(), 14))) # Michael S. Branicky, Apr 12 2024

a(15) from Daniel Suteu, Dec 17 2024

A165234 Least prime p such that 2x^2 + p produces primes for x=0..n-1 and composite for x=n.

Original entry on oeis.org

2, 17, 3, 1481, 5, 149, 569, 2081, 2339, 5939831, 11, 33164857769, 3217755097229, 272259344081, 17762917045631

Offset: 1

T. D. Noe, Sep 09 2009

Other known values: a(14)=272259344081 and a(29)=29. There are no other terms less than 10^12. The primes p = 3, 5, 11, and 29 produce p consecutive distinct primes because the imaginary quadratic field Q(sqrt(-2p)) has class number 2. Assuming the prime k-tuples conjecture, this sequence is defined for n>0.

Paulo Ribenboim, My Numbers, My Friends, Springer,2000, pp. 349-350.

R. A. Mollin, Prime-producing quadratics, Amer. Math. Monthly 104 (1997), 529-544.
Eric W. Weisstein, Prime-Generating Polynomial

Cf. A007641, A050265, A161008, A164926.

PrimeRun[p_Integer] := Module[{k=0}, While[PrimeQ[2k^2+p], k++ ]; k]; nn=9; t=Table[0,{nn}]; cnt=0; p=1; While[cnt

isok(p, n) = for (k=0, n-1, if(!isprime(p + 2*k^2), return(0))); return(!isprime(p + 2*n^2));
a(n) = forprime(p=2, oo, if(isok(p, n), return(p))); \\ Daniel Suteu, Dec 22 2024

use ntheory qw(:all); sub a { my $n = $[0]; my $lo = 2; my $hi = 2*$lo; while (1) { my @terms = grep { !is_prime($ + 2*$n*$n) } sieve_prime_cluster($lo, $hi, map { 2*$*$ } 1 .. $n-1); return $terms[0] if @terms; $lo = $hi+1; $hi = 2*$lo; } }; $| = 1; for my $n (1..100) { print a($n), ", " } # Daniel Suteu, Dec 22 2024

a(13) and a(15) from Daniel Suteu, Dec 22 2024

cp's OEIS Frontend

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

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A210360 Prime numbers p such that x^2 + x + p produces primes for x = 0..1 but not x = 2.

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A210361 Prime numbers p such that x^2 + x + p produces primes for x = 0..2 but not x = 3.

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A210362 Prime numbers p such that x^2 + x + p produces primes for x = 0..3 but not x = 4.

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A210363 Prime numbers p such that x^2 + x + p produces primes for x = 0..4 but not x = 5.

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Mathematica

A210364 Prime numbers p such that x^2 + x + p produces primes for x = 0..5 but not x = 6.

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Mathematica

A210365 Prime numbers p such that x^2 + x + p produces primes for x = 0..6 but not x = 7.

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Mathematica

A370387 a(n) is the least prime p such that p + 6*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.

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A371024 a(n) is the least prime p such that p + 4*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.

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A370387 a(n) is the least prime p such that p + 6k(k+1) is prime for 0 <= k <= n-1 but not for k=n.

A371024 a(n) is the least prime p such that p + 4k(k+1) is prime for 0 <= k <= n-1 but not for k=n.