A007635 -id:A007635 - cp's OEIS frontend

A272285 Primes of the form 43n^2 - 537n + 2971 in order of increasing nonnegative values of n.

2971, 2477, 2069, 1747, 1511, 1361, 1297, 1319, 1427, 1621, 1901, 2267, 2719, 3257, 3881, 4591, 5387, 6269, 7237, 8291, 9431, 10657, 11969, 13367, 14851, 16421, 18077, 19819, 21647, 23561, 25561, 27647, 29819, 32077, 34421, 39367, 41969, 44657, 47431, 50291

Offset: 1

Robert Price, Apr 24 2016

			1511 is in this sequence since 43*4^2 - 537*4 + 2971 = 688-2148+2971 = 1511 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272118, A272159, A271143, A272284.

n = Range[0, 100]; Select[43n^2 - 537n + 2971, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(p=43*n^2 - 537*n + 2971), print1(p, ", "))); \\ Altug Alkan, Apr 24 2016

A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

Original entry on oeis.org

1, 11, 17, 41, 21377, 27941, 41537, 55661, 115721, 239621, 247757

Offset: 1

Hugo Pfoertner, Feb 02 2020

The Hardy and Littlewood Conjecture F provides an estimate of the number of primes generated by a quadratic polynomial P(x) for 0 <= x <= m in the form C * Integral_{x=2..m} 1/log(x) dx, with C given by an Euler product that is a function of the fundamental discriminant of the polynomial. Cohen describes an efficient method for the computation of C.
The following table provides the record values of C, together with the number of primes np generated by the polynomial x^2 + x + a(n) for x <= 10^8 and the actual ratio 2*np/Integral_{x=2..10^8} 1/log(x) dx.
    a(n)    C       np    C from ratio
       1 2.24147  6456835 2.24110
      11 3.25944  9389795 3.25910
      17 4.17466 12027453 4.17460
      41 6.63955 19132653 6.64073
   21377 6.92868 19962992 6.92894
   27941 7.26400 20931145 7.26497
   41537 7.32220 21092134 7.32085
   55661 7.45791 21483365 7.45664
  115721 7.70935 22210771 7.70912
  239621 7.72932 22268336 7.72909
  247757 8.24741 23762118 8.24757
Jacobson and Williams found significantly larger values of C for very large addends k, e.g. C = 2*5.36708 = 10.73416 for k = 3399714628553118047.

Keith Conrad, Hardy-Littlewood Constants. In: Mathematical Properties of Sequences and Other Combinatorial Structures, eds. Jong-Seon No, Hong-Yeop Song, Tor Helleseth, P. Vijay Kumar, Springer New York, 2003, pages 133-154.

Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).
Michael J. Jacobson Jr. and Hugh C. Williams, New Quadratic Polynomials With High Densities Of Prime Values, Math. Comp., 72, 241, 499-519, 2002.

Cf. A002837, A007635, A014556, A116206, A331877.

Cf. A221712, A221713 (Constants C including factor 1/2).

\\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2max=0; for(add=0,100,my(hl=HardyLittlewood2(n^2+n+add)); if(hl>hl2max,print1(add,", "); hl2max=hl))

A272284 Numbers n such that 43n^2 - 537n + 2971 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, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 55, 56, 57, 60, 64, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81

Offset: 1

Robert Price, Apr 24 2016

nonn

35 is the smallest number not in this sequence.

			4 is in this sequence since 43*4^2 - 537*4 + 2971 = 688-2148+2971 = 1511 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000
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.

Select[Range[0, 100], PrimeQ[43#^2 - 537# + 2971] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(43*n^2 - 537*n + 2971), print1(n, ", "))); \\ Altug Alkan, Apr 24 2016

A272401 Primes of the form abs(3n^3 - 183n^2 + 3318n - 18757) in order of increasing nonnegative n.

Original entry on oeis.org

18757, 15619, 12829, 10369, 8221, 6367, 4789, 3469, 2389, 1531, 877, 409, 109, 41, 59, 37, 229, 499, 829, 1201, 1597, 1999, 2389, 2749, 3061, 3307, 3469, 3529, 3469, 3271, 2917, 2389, 1669, 739, 419, 1823, 3491, 5441, 7691, 10259, 13163, 16421, 20051, 24071

Offset: 1

Robert Price, Apr 28 2016

			8221 is in this sequence since abs(3*4^3 - 183*4^2 + 3318*4 - 18757) = abs(192-2928+13272-18757) = 8221 is prime.

Robert Price, Table of n, a(n) for n = 1..2839
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272159, A271143, A272284, A272302.

n = Range[0, 100]; Select[3n^3 - 183n^2 + 3318n - 18757 , PrimeQ[#] &]

A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

Original entry on oeis.org

2, 3, 5, 5, 7, 11, 7, 9, 13, 19, 11, 13, 17, 23, 31, 13, 15, 19, 25, 33, 43, 17, 19, 23, 29, 37, 47, 59, 19, 21, 25, 31, 39, 49, 61, 75, 23, 25, 29, 35, 43, 53, 65, 79, 95, 29, 31, 35, 41, 49, 59, 71, 85, 101, 119, 31, 33, 37, 43, 51, 61, 73, 87, 103, 121, 141, 37, 39, 43, 49, 57

Offset: 1

Reinhard Zumkeller, Mar 25 2006

A117531 gives the number of primes in the n-th row;

if T(n,1) is a Lucky Number of Euler then A117531(n)=n, see A014556.

			T(5,k)=A048058(k)=A048059(k), 1<=k<=5: T(5,1)=A014556(4)=11;
T(7,k)=A007635(k), 1<=k<=7: T(7,1)=A014556(5)=17;
T(13,k)=A005846(k), 1<=k<=13: T(13,1)=A014556(6)=41.

Robert Price, Table of n, a(n) for n = 1..10011
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Eric Weisstein's World of Mathematics, Lucky Number of Euler

Cf. A005846, A007635, A014556, A048058, A048059, A117531.

Table[k^2 - k + Prime[n], {n, 141}, {k, n}] // Flatten (* Robert Price, Apr 19 2025 *)

T(n,k) = k^2 - k + prime(n) \\ Charles R Greathouse IV, Apr 24 2015

T(n,1) = A000040(k).
T(n,2) = A052147(k) for k>1.
For 1

A272118 Numbers k such that abs(6k^2 - 342k + 4903) 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, 61, 62, 64, 66, 67, 68, 69, 71, 72

Offset: 1

Robert Price, Apr 20 2016

nonn

			4 is in this sequence since 6*4^2 - 342*4 + 4903 = 96-1368+4903 = 3631 is prime.

Robert Price, Table of n, a(n) for n = 1..3874
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074, A272075.

Select[Range[0, 100], PrimeQ[6*#^2 - 342*# + 4903] &]

isok(n) = isprime(abs(6*n^2 - 342*n + 4903)); \\ Michel Marcus, Apr 21 2016

A272302 Nonnegative numbers n such that abs(3n^3 - 183n^2 + 3318n - 18757) 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, 48, 49, 51, 53, 56, 57, 59, 60, 62, 63, 65, 66, 69, 70, 74, 79, 80, 81, 82, 85

Offset: 1

Robert Price, Apr 28 2016

47 is the smallest number not in this sequence.

			4 is in this sequence since abs(3*4^3 - 183*4^2 + 3318*4 - 18757) = abs(192-2928+13272-18757) = 8221 is prime.

Robert Price, Table of n, a(n) for n = 1..2839
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266, A271980, A272030, A272074, A272075, A272160, A271144, A272285, A272401.

Select[Range[0, 100], PrimeQ[3#^3 - 183#^2 + 3318# - 18757 ] &]

is(n)=isprime(abs(3*n^2-183*n^2+3318*n-18757)) \\ Charles R Greathouse IV, Feb 17 2017

A272438 Primes of the form abs(-66n^3 + 3845n^2 - 60897n + 251831) in order of increasing nonnegative n.

Original entry on oeis.org

251831, 194713, 144889, 101963, 65539, 35221, 10613, 8681, 23057, 32911, 38639, 40637, 39301, 35027, 28211, 19249, 8537, 3529, 16553, 30139, 43891, 57413, 70309, 82183, 92639, 101281, 107713, 111539, 112363, 109789, 103421, 92863, 77719, 57593, 32089, 811

Offset: 1

Robert Price, Apr 29 2016

nonn

			65539 is in this sequence since abs(-66*4^3 + 3845*4^2 - 60897*4 + 251831) = abs(-4224+61520-243588+251831) = 65539 is prime.

Robert Price, Table of n, a(n) for n = 1..3012
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272159, A271143, A272284, A272302, A272437.

n = Range[0, 100]; Select[-66n^3 + 3845n^2 - 60897n + 251831, PrimeQ[#] &]

A272437 Nonnegative numbers n such that abs(-66n^3 + 3845n^2 - 60897n + 251831) 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, 47, 49, 51, 54, 58, 65, 68, 70, 75, 76, 77, 82, 88, 89, 97, 99, 101, 102, 104, 109

Offset: 1

Robert Price, Apr 29 2016

nonn

46 is the smallest number not in this sequence.

			4 is in this sequence since abs(-66*4^3 + 3845*4^2 - 60897*4 + 251831) = abs(-4224+61520-243588+251831) = 65539 is prime.

Robert Price, Table of n, a(n) for n = 1..3012
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266, A271980, A272030, A272074, A272075, A272160, A271144, A272285, A272401, A272438.

Select[Range[0, 109], PrimeQ[-66#^3 + 3845#^2 - 60897# + 251831] &]

is(n)=isprime(abs(66*n^3-3845*n^2+60897*n-251831)) \\ Charles R Greathouse IV, Feb 20 2017

A272444 Primes of the form abs(n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397) in order of increasing nonnegative n.

Original entry on oeis.org

286397, 8543, 210011, 336121, 402851, 424163, 412123, 377021, 327491, 270631, 212123, 156353, 106531, 64811, 32411, 9733, 3517, 8209, 5669, 2441, 14243, 27763, 41051, 52301, 59971, 62903, 60443, 52561, 39971, 24251, 7963, 5227, 10429, 1409, 29531, 91673

Offset: 1

Robert Price, Apr 29 2016

nonn

			402851 is in this sequence since abs(4^5 - 99*4^4 + 3588*4^3 - 56822*4^2 + 348272*4 - 286397) = abs(1024-25344+229632-909152+1393088-286397) = 402851 is prime.

Robert Price, Table of n, a(n) for n = 1..2170
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272030, A272074, A272075, A272159, A271143, A272284, A272302, A272437, A272443.

n = Range[0, 100]; Select[n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(n^5-99*n^4+3588*n^3-56822*n^2+348272*n-286397)), print1(p, ", "))); \\ Altug Alkan, Apr 29 2016

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A272285 Primes of the form 43*n^2 - 537*n + 2971 in order of increasing nonnegative values of n.

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A331940 Addends k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant.

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A272284 Numbers n such that 43*n^2 - 537*n + 2971 is prime.

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A272401 Primes of the form abs(3n^3 - 183n^2 + 3318n - 18757) in order of increasing nonnegative n.

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A117530 Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.

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A272118 Numbers k such that abs(6*k^2 - 342*k + 4903) is prime.

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A272302 Nonnegative numbers n such that abs(3n^3 - 183n^2 + 3318n - 18757) is prime.

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A272438 Primes of the form abs(-66n^3 + 3845n^2 - 60897n + 251831) in order of increasing nonnegative n.

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A272285 Primes of the form 43n^2 - 537n + 2971 in order of increasing nonnegative values of n.

A272284 Numbers n such that 43n^2 - 537n + 2971 is prime.

A272118 Numbers k such that abs(6k^2 - 342k + 4903) is prime.