A050267 -id:A050267 - cp's OEIS frontend

A272076 Numbers n such that abs(7n^2 - 371n + 4871) is prime.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 59, 61, 63, 65, 67, 68, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Robert Price, Apr 19 2016

nonn

			4 is in this sequence since 7*4^2 - 371*4 + 4871 = 112-1484+4871 = 3499 is prime.

Robert Price, Table of n, a(n) for n = 1..3530
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272074, A272075, A272077.

Select[Range[0, 100], PrimeQ[7#^2 - 371# + 4871] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(abs(7*n^2-371*n+4871)), print1(n, ", "))); \\ Altug Alkan, Apr 19 2016

A272813 Nonnegative numbers n such that n^2 - 79n + 1601 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset: 1

Robert Price, May 06 2016

nonn

80 is the smallest number not in this sequence.

See A005846 for the corresponding primes.

			4 is in this sequence since 4^2 - 79*4 + 1601 = 16-316+1601 = 1301 is prime.

Robert Price, Table of n, a(n) for n = 1..4179
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272030, A272074, A272075, A272160, A271144, A272285, A272401, A272438, A272444, A272410, A272555, A272710, A005846.

Select[Range[0, 100], PrimeQ[#^2 - 79# + 1601 ] &]

lista(nn) = for(n=0, nn, if(isprime(n^2-79*n+1601), print1(n, ", "))); \\ Altug Alkan, May 06 2016

A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 73, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 89, 91, 97, 100

Offset: 1

James R. Buddenhagen, Mar 06 2018

nonn

The quadratic polynomial p(k) = k^2 + 21*k + 1 is not a prime-generating polynomial in the sense of Eric Weisstein's World of Mathematics (see link) because p(0) is not prime.

However p(k) is prime for the first 17 positive integral values of k and among polynomials of the form k^2 + j*k + 1, the present polynomial (j = 21) generates more primes than any polynomial of that form for any positive integral j < 231, at least for positive integers, k, in the range 0 < k < 10^6.

			17 is in the sequence because 17^2 + 21 * 17 + 1 = 647 is prime.
18 is not in the sequence because 18^2 + 21 * 18 + 1 = 703 = 19 * 37.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A292509, A050267, A050268, A005846, A007635, A007641, A048988.

select(k-> isprime(k^2+21*k+1), [$1..100])

Select[Range[100], PrimeQ[#^2 + 21# + 1] &] (* Alonso del Arte, Mar 06 2018 *)

isok(k) = isprime(k^2+21*k+1); \\ Altug Alkan, Mar 07 2018

A253239 Numbers k such that k^2 + k + 72491 is prime.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 53, 55, 56, 57, 58, 59, 64, 65, 66, 67, 72, 73, 74, 75, 77, 78, 81, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 98, 100

Offset: 1

Eric Chen, Apr 19 2015

Of the first 10000 natural numbers, 4534 are in this sequence, making the density about 45%, quite large! (However, 72491 is not prime; it equals 71*1021, so no multiples of 71 or 1021 are in this sequence.)

			k       k^2 + k + 72491
0       72491 = 71*1021
1       72493 (prime)
2       72497 (prime)
3       72503 (prime)
4       72511 = 59*1229
5       72521 = 47*1543
6       72533 (prime)
7       72547 (prime)
8       72563 = 149*487
9       72581 = 181*401
etc.

Eric Chen, Table of n, a(n) for n = 1..4534 (all terms up to 10000)
C. Rivera, Prime producing polynomials
Eric Weisstein's World of Mathematics, Prime-generating polynomial

Cf. A048097, A028823, A056561, A141489, A160548, A116206, A050267, A128878.

[n: n in [0..100] | IsPrime(n^2 + n + 72491)]; // Vincenzo Librandi, Apr 20 2015

select(t -> isprime(t^2+t+72491), [$0..100]);

Select[Range[100], PrimeQ[#^2 + # + 72491] &]

v=[ ]; for(n=0, 100, if(isprime(n^2+n+72491), v=concat(v, n), )); v

A301985 a(n) = n^2 + 2329*n + 1697.

Original entry on oeis.org

1697, 4027, 6359, 8693, 11029, 13367, 15707, 18049, 20393, 22739, 25087, 27437, 29789, 32143, 34499, 36857, 39217, 41579, 43943, 46309, 48677, 51047, 53419, 55793, 58169, 60547, 62927, 65309, 67693, 70079, 72467, 74857, 77249, 79643, 82039, 84437, 86837, 89239, 91643, 94049, 96457, 98867, 101279

Offset: 0

Dmitry Kamenetsky, Mar 30 2018

This quadratic seems to be good at generating many distinct primes. It generates 642 primes for n < 10^3, 5132 primes for n < 10^4, 41224 primes for n < 10^5 and 340009 primes for n < 10^6. The first few primes generated are 1697, 4027, 6359, 8693, 13367, 18049, 20393, 22739, 25087, 27437.
The quadratic was first discovered as b(n) = n^2 + 1151n - 1023163 by Steve Trevorrow in 2006 during Al Zimmermann's Prime Generating Polynomials contest (see link). Note that a(n) = b(n + 589).
Smallest n such that a(n) is not squarefree is 704; a(704) = 2136929 = 73^2*401 and smallest n such that a(n) is square is 1327; a(1327) = 4853209 = 2203^2. - Altug Alkan, Mar 30 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Al Zimmermann, Prime Generating Polynomials contest, 2006.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005846, A050267, A050268.

[n^2+2329*n+1697: n in [0..50]]; // Vincenzo Librandi, Mar 31 2018

Table[n^2 + 2329 n + 1697, {n, 0, 50}] (* Vincenzo Librandi, Mar 31 2018 *)

a(n) = n^2 + 2329*n + 1697; \\ Altug Alkan, Mar 30 2018

Vec((1697 - 1064*x - 631*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Mar 30 2018

a(n) = 2*a(n-1) - a(n-2) + 2, a(0) = 1697, a(1) = 4027.
From Colin Barker, Mar 30 2018: (Start)
G.f.: (1697 - 1064*x - 631*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
E.g.f.: exp(x)*(1697 + 2330*x + x^2). - Elmo R. Oliveira, Feb 10 2025

A217604 Primes or negative values of primes of the form 59n^2 - 1873n + 8941 for n>=0.

Original entry on oeis.org

8941, 7127, 5431, 3853, 2393, 1051, -173, -1279, -2267, -3137, -3889, -4523, -5039, -5437, -5717, -5879, -5923, -5849, -5657, -5347, -4919, -4373, -3709, -2927, -2027, -1009, 127, 1381, 2753, 4243, 5851, 7577, 9421, 11383, 13463, 15661, 17977, 20411, 22963, 25633, 31327, 34351

Offset: 1

Pedja Terzic, Oct 08 2012

Terms are listed in the order of appearance. The absolute values are primes for 0 <= n <= 39.

Eric Weisstein's World of Mathematics, Prime Generating Polynomials

Cf. A050267, A050268, A217439, A217440.

Select[Table[59*n^2-1873*n+8941,{n, 0, 50}], PrimeQ[#]&]

[n | n <- apply(m->59*m^2-1873*m+8941, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

cp's OEIS Frontend

A272076 Numbers n such that abs(7*n^2 - 371*n + 4871) is prime.

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A272813 Nonnegative numbers n such that n^2 - 79n + 1601 is prime.

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A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

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A253239 Numbers k such that k^2 + k + 72491 is prime.

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A301985 a(n) = n^2 + 2329*n + 1697.

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A217604 Primes or negative values of primes of the form 59*n^2 - 1873*n + 8941 for n>=0.

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A272076 Numbers n such that abs(7n^2 - 371n + 4871) is prime.

A217604 Primes or negative values of primes of the form 59n^2 - 1873n + 8941 for n>=0.