A050267 -id:A050267 - cp's OEIS frontend

A272077 Primes of the form abs(7k^2 - 371k + 4871) in order of increasing nonnegative values of k.

4871, 4507, 4157, 3821, 3499, 3191, 2897, 2617, 2351, 2099, 1861, 1637, 1427, 1231, 1049, 881, 727, 587, 461, 349, 251, 167, 97, 41, 29, 43, 43, 29, 41, 97, 167, 251, 349, 461, 587, 727, 881, 1049, 1231, 1427, 1637, 1861, 2099, 2351, 2617, 2897, 3191

Offset: 1

Robert Price, Apr 19 2016

nonn

For k=0 to 23, this expression generates 24 primes that decrease from 4871 to 41. It generates duplicates and the absolute value is used to avoid negative terms. The same 24 primes but in reverse order are generated in the same range of the argument by 7*k^2+49*k+41, which produces neither duplicates nor negative values and is one of relatively few quadratics with at most two-digit coefficients that generate at least 20 primes in a row. We have: 7*(n-30)^2 + 49*(n-30) + 41 = 7*n^2 - 371*n + 4871. - Waldemar Puszkarz, Feb 02 2018

See also A298078, the values of 7*n^2-7*n-43, which also contains the same 24 primes without duplicates. - N. J. A. Sloane, Mar 10 2018

			4157 is in this sequence since 7*2^2 - 371*2 + 4871 = 28-742-4871 = 4157 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, A272076, A298078.

Filtered(List([0..10^2],n->7*n^2-371*n+4871),IsPrime); # Muniru A Asiru, Feb 04 2018

select(isprime, [seq(7*n^2-371*n+4871, n=0..10^2)]); # Muniru A Asiru, Feb 04 2018

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

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

A272324 Primes of the form abs(82n^3 - 1228n^2 + 6130n - 5861) in order of increasing nonnegative n.

Original entry on oeis.org

5861, 877, 2143, 3691, 4259, 4339, 4423, 5003, 6571, 9619, 14639, 22123, 32563, 46451, 64279, 86539, 113723, 146323, 184831, 229739, 281539, 340723, 407783, 483211, 567499, 661139, 764623, 878443, 1003091, 1139059, 1286839, 1446923, 2005919, 2693363, 3229579

Offset: 1

Robert Price, Apr 25 2016

nonn

			4259 is in this sequence since 82*4^3 - 1228*4^2 + 6130*4 - 5861 = 5248-19648+24520-5861 = 4259 is prime.

Robert Price, Table of n, a(n) for n = 1..2659
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A076808, A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272030, A272074, A272075, A272118, A272159, A271143, A272284, A272323.

n = Range[0, 100]; Select[82n^3 - 1228n^2 + 6130n - 5861, PrimeQ[#] &]

A272325 Nonnegative numbers n such that n^4 + 853n^3 + 2636n^2 + 3536n + 1753 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, 22, 25, 26, 27, 30, 34, 37, 41, 43, 46, 50, 52, 53, 56, 59, 60, 61, 64, 66, 67, 68, 71, 76, 79, 81, 84, 87, 88, 89, 91, 92, 95, 96, 98, 99, 103, 106, 109, 118, 124, 126, 127, 128, 132

Offset: 1

Robert Price, Apr 25 2016

nonn

21 is the smallest number not in this sequence.

			4 is in this sequence since 4^4 + 853*4^3 + 2636*4^2 + 3536*4 + 1753 = 256+54592+42176+14144+1753 = 112921 is prime.

Robert Price, Table of n, a(n) for n = 1..2457
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, A272326, A076809.

Select[Range[0, 100], PrimeQ[#^4 + 853#^3 + 2636#^2 + 3536# + 1753] &]

lista(nn) = for(n=0, nn, if(isprime(n^4+853*n^3+2636*n^2+3536*n+1753), print1(n, ", "))); \\ Altug Alkan, Apr 25 2016

A272326 Primes of the form k^4 + 853k^3 + 2636k^2 + 3536*k + 1753 in order of increasing nonnegative k.

Original entry on oeis.org

1753, 8779, 26209, 59197, 112921, 192583, 303409, 450649, 639577, 875491, 1163713, 1509589, 1918489, 2395807, 2946961, 3577393, 4292569, 5097979, 5999137, 7001581, 8110873, 10672369, 15456403, 17324929, 19339909, 26321233, 38031841, 48822439, 66193219

Offset: 1

Robert Price, Apr 25 2016

nonn

			112921 is in this sequence since 4^4 + 853*4^3 + 2636*4^2 + 3536*4 + 1753 = 256+54592+42176+14144+1753 = 112921 is prime.

Robert Price, Table of n, a(n) for n = 1..2457
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, A272323, A272325, A076809.

n = Range[0, 100]; Select[n^4 + 853n^3 + 2636n^2 + 3536n + 1753, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=n^4+853*n^3+2636*n^2+3536*n+1753), print1(p, ", "))); \\ Altug Alkan, Apr 25 2016

A272554 Nonnegative numbers n such that abs(1/(36)(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236)) 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, 57, 61, 62, 63, 64, 65, 66, 68, 69, 70, 73, 78

Offset: 1

Robert Price, May 02 2016

nonn

55 is the smallest number not in this sequence.

			4 is in this sequence since abs(1/(36)(4^6 - 126*4^5 + 6217*4^4 - 153066*4^3 + 1987786*4^2 - 13055316*4 + 34747236)) = abs((4096 - 129024 + 1591552 - 9796224 + 31804576 - 5222126 + 34747236)/36) = 166693 is prime.

Robert Price, Table of n, a(n) for n = 1..2128
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.

Select[Range[0, 100], PrimeQ[1/(36)(#^6 - 126#^5 + 6217#^4 - 153066#^3 + 1987786#^2 - 13055316# + 34747236)] &]

A272710 Primes of the form abs((1/4)*(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) in order of increasing nonnegative n.

Original entry on oeis.org

1705829, 1313701, 991127, 729173, 519643, 355049, 228581, 134077, 65993, 19373, 10181, 26539, 33073, 32687, 27847, 20611, 12659, 5323, 383, 3733, 4259, 1721, 3923, 12547, 23887, 37571, 53149, 70123, 87977, 106207, 124351, 142019, 158923, 174907, 189977

Offset: 1

Robert Price, May 04 2016

nonn

			519643 is in this sequence since abs(1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) = abs((1024 - 34048 + 430656 - 2534064 + 6881176 - 6823316)/4) = 519643 is prime.

Robert Price, Table of n, a(n) for n = 1..2328
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, A268200, A272554, A247163.

n = Range[0, 100]; Select[1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316), PrimeQ[#] &]

A105551 Number of distinct prime factors of n^3 + n^2 + 71.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 3, 3, 2, 1, 2, 1, 1, 3, 2, 3, 1, 2, 1, 1, 3, 2, 1, 2, 3, 2, 2, 2, 1, 3, 1, 1, 3, 2, 1, 2

Offset: 0

Jonathan Vos Post, May 03 2005

This cubic equation with small positive coefficients is strangely rich in primes and semiprimes. The first 44 consecutive values, for n = 0, 1, 2, ..., 43, are all either prime (23 of them) or semiprime (21 of them), before the first 3-almost prime value is encountered.

			a(0) = 1 because 0^3 + 0^2 + 71 = 71 is prime.
a(1) = 1 because 1^3 + 1^2 + 71 = 73 is prime.
a(2) = 1 because 2^3 + 2^2 + 71 = 83 is prime.
a(3) = 1 because 3^3 + 3^2 + 71 = 107 is prime.
a(4) = 1 because 3^3 + 3^2 + 71 = 151 is prime.
a(5) = 2 because 3^3 + 3^2 + 71 = 221 = 13 * 17 is the first semiprime.
a(44) = 3 because 44^3 + 44^2 + 71 = 87191 = 13 * 19 * 353 is the first 3-almost prime for nonnegative integers n.

T. D. Noe, Table of n, a(n) for n = 0..1000
Ulrich Abel and Hartmut Siebert, Sequences with Large Numbers of Prime Values, Am. Math. Monthly 100, 167-169, 1993.
Robin Forman, Sequences with Many Primes, Amer. Math. Monthly 99, 548-557, 1992.
Betty Garrison, Polynomials with Large Numbers of Prime Values, Amer. Math. Monthly 97, 316-317, 1990.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A000040, A001358, A005846, A007635, A007641, A048988, A050265, A050268, A050267, A050266.

Table[PrimeNu[n^3+n^2+71],{n,0,90}] (* Harvey P. Dale, Oct 09 2012 *)

a(n)=omega(n^3 + n^2 + 71) \\ Charles R Greathouse IV, Jan 31 2017

a(n) = A001221(n^3 + n^2 + 71).

More terms from Robert G. Wilson v, May 21 2005

A128878 Primes of the form 47n^2 - 1701n + 10181.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387, 34057, 36821, 39679, 45677, 48817, 52051, 65927, 81307, 89561, 102647, 107197, 116579, 126337, 131357

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Apr 17 2007

nonn

Primes are given in the order in which they arise for increasing n.
Polynomial generates 22 primes for 0 <= n <= 42, i.e., for n = 0, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42.
If the definition is replaced by "Numbers n of the form 47*k^2 - 1701*k + 10181 such that either n or -n is a prime" we get (essentially) A050267.

			47k^2 - 1701k + 10181 = 21647 for k = 42.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, ISBN 0-387-20860-7, Section A17, page 59.

G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles and Problems Connection.

Cf. A050267, A002383, A027753, A027755, A005471, A027758, A048059, A007635, A005846, A116206, A050268, A022464.

Select[Table[47*n^2 - 1701*n + 10181, {n, 0, 100}], # > 0 && PrimeQ[#] &] (* T. D. Noe, Aug 02 2011 *)

Edited by Klaus Brockhaus, Apr 22 2007 and by N. J. A. Sloane, May 05 2007 and May 06 2007

A247163 Nonnegative numbers n such that abs(1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) 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, 59, 60, 61, 64, 67, 68, 69, 74, 75, 76

Offset: 1

Robert Price, May 04 2016

nonn

62 is the smallest number not in this sequence.

			4 is in this sequence since abs(1/4 (n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) = abs((1024 - 34048 + 430656 - 2534064 + 6881176 - 6823316)/4) = 519643 is prime.

Robert Price, Table of n, a(n) for n = 1..2328
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.

Select[Range[0, 100], PrimeQ[1/4 (#^5 - 133#^4 + 6729#^3 - 158379#^2 + 1720294# - 6823316)] &]

A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) 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, 44, 45, 47, 49, 50, 51, 52, 53, 54, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 73, 74, 76, 77, 80

Offset: 1

Robert Price, Apr 28 2016

nonn

43 is the smallest number not in this sequence.

See A267252 for more information. - Hugo Pfoertner, Dec 13 2019

			4 is in this sequence since 103*4^2 - 4707*4 + 50383  = 1648-18828+50383 = 33203 is prime.

Robert Price, Table of n, a(n) for n = 1..3885
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, A267252.

Select[Range[0, 100], PrimeQ[103#^2 - 4707# + 50383 ] &]

lista(nn) = for(n=0, nn, if(isprime(abs(103*n^2-4707*n+50383)), print1(n, ", "))); \\ Altug Alkan, Apr 28 2016, corrected by Hugo Pfoertner, Dec 13 2019

Title corrected by Hugo Pfoertner, Dec 13 2019

cp's OEIS Frontend

A272077 Primes of the form abs(7*k^2 - 371*k + 4871) in order of increasing nonnegative values of k.

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A272324 Primes of the form abs(82n^3 - 1228n^2 + 6130n - 5861) in order of increasing nonnegative n.

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A272325 Nonnegative numbers n such that n^4 + 853n^3 + 2636n^2 + 3536n + 1753 is prime.

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A272326 Primes of the form k^4 + 853*k^3 + 2636*k^2 + 3536*k + 1753 in order of increasing nonnegative k.

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A272554 Nonnegative numbers n such that abs(1/(36)(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236)) is prime.

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A272710 Primes of the form abs((1/4)*(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294n - 6823316)) in order of increasing nonnegative n.

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A105551 Number of distinct prime factors of n^3 + n^2 + 71.

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A128878 Primes of the form 47*n^2 - 1701*n + 10181.

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A272077 Primes of the form abs(7k^2 - 371k + 4871) in order of increasing nonnegative values of k.

A272326 Primes of the form k^4 + 853k^3 + 2636k^2 + 3536*k + 1753 in order of increasing nonnegative k.

A128878 Primes of the form 47n^2 - 1701n + 10181.

A267069 Nonnegative numbers n such that abs(103n^2 - 4707n + 50383) is prime.