A271980 -id:A271980 - cp's OEIS frontend

A048988 Primes of the form 4k^2 + 4k + 59.

59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499, 587, 683, 787, 1019, 1283, 1427, 1579, 1907, 2083, 2267, 2459, 2659, 3083, 3307, 3539, 3779, 4027, 4283, 4547, 5099, 5387, 5683, 5987, 6299, 6619, 6947, 7283, 8707, 9467, 9859, 10259, 10667, 11083

Offset: 1

G. L. Honaker, Jr.

From Peter Bala, Apr 18 2018: (Start)
Let P(n) = 4*n^2 + 4*n + 59. The polynomial 1/2*P(n-1/2) = 2*n^2 + 29 has prime values for n from 0 to 28. See A007641. Also P(n-14) = 4*n^2 - 108*n + 787 is prime for the 28 consecutive values of n from 0 to 27.
The sequence of 28 values of the polynomial 4*P((n-2)/4) = n^2 + 232 for n from -1 to 26 is [233, 2^3*29, 233, 2^2*59, 241, 2^3*31, 257, 2^2*67, 281, 2^3*37, 313, 2^2*83, 353, 2^3*47, 401, 2^2*107, 457, 2^3*61, 521, 2^2*139, 593, 2^3*79, 673, 2^2*179, 761, 2^3*101, 857, 2^2*227], and consists of 7 groups of 4 numbers of the form p_1, 2^3*p_2, p_3, 2^2*p_4, where the p's are prime numbers. (End)

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

Cf. A005846, A007641, A271980.

[ a: n in [0..250] | IsPrime(a) where a is 4*n^2 +4*n + 59]; // Vincenzo Librandi, Nov 19 2010

select(isprime, [4*k*(k+1)+59$k=0..100])[];  # Alois P. Heinz, Apr 16 2025

Select[(4 #^2 + 4 # + 59) & /@ Range[0, 100], PrimeQ] (* Robert Price, Apr 16 2025 *)

lista(nn) = for(k=0, nn, if(isprime(p=4*k^2+4*k+59), print1(p, ", "))); \\ Altug Alkan, Apr 18 2018

A272074 Numbers k such that k^4 + 29*k^2 + 101 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, 21, 23, 26, 31, 32, 34, 35, 37, 43, 44, 45, 47, 49, 53, 56, 60, 61, 62, 66, 67, 68, 70, 71, 72, 74, 75, 79, 80, 81, 84, 85, 89, 90, 91, 93, 96, 99

Offset: 1

Robert Price, Apr 19 2016

			4 is in this sequence since 4^4 + 29*4^2 + 101 = 256+464+101 = 821 is prime.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585.

Cf. A271980, A272075.

Select[Range[0,100],PrimeQ[#^4+29#^2+101]&] (* Harvey P. Dale, Dec 15 2020 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(n^4+29*n^2+101), print1(n, ", "))); \\ Altug Alkan, Apr 19 2016

A272075 Primes of the form k^4 + 29*k^2 + 101.

Original entry on oeis.org

101, 131, 233, 443, 821, 1451, 2441, 3923, 6053, 9011, 13001, 18251, 25013, 33563, 44201, 57251, 73061, 92003, 114473, 140891, 207371, 295283, 476681, 951491, 1078373, 1369961, 1536251, 1913963, 3472523, 3804341, 4159451, 4943843, 5834531, 7972043, 9925541

Offset: 1

Robert Price, Apr 19 2016

			233 is prime and it is in this sequence since 233 = 2^4 + 29*2^2 + 101.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074.

n = Range[0, 100]; Select[#^4 + 29#^2 + 101, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(p=n^4+29*n^2+101), print1(p, ", "))); \\ Altug Alkan, Apr 19 2016

A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

Original entry on oeis.org

7243, 6763, 6299, 5851, 5419, 5003, 4603, 4219, 3851, 3499, 3163, 2843, 2539, 2251, 1979, 1723, 1483, 1259, 1051, 859, 683, 523, 379, 251, 139, 43, 37, 101, 149, 181, 197, 197, 181, 149, 101, 37, 43, 139, 251, 379, 523, 683, 859, 1051, 1259, 1483, 1723, 1979

Offset: 1

Robert Price, Apr 21 2016

nonn

			5419 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

Robert Price, Table of n, a(n) for n = 1..4058
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

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

Cf. A271980, A272074, A272075, A272118, A272159.

n = Range[0, 100]; Select[Abs[8n^2 - 488n + 7243], PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(isprime(p=abs(8*n^2-488*n+7243)), print1(p, ", "))); \\ Altug Alkan, Apr 21 2016

A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

Original entry on oeis.org

250703, 224579, 199247, 174959, 151967, 130523, 110879, 93287, 77999, 65267, 55343, 48479, 44927, 44939, 48767, 56663, 68879, 85667, 107279, 133967, 165983, 203579, 247007, 296519, 352367, 414803, 484079, 560447, 644159, 735467, 834623, 941879, 1057487

Offset: 1

Robert Price, Apr 23 2016

nonn

			151967 is prime and it is in this sequence since 151967 = 42*4^3 + 270*4^2 - 26436*4 + 250703.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265 - A050268, A005846, A007641, A007635, A048988, A256585

Cf. A271980, A272074, A272075, A272118, A272159, A271143 (associated k).

n = Range[0, 100]; Select[42n^3 + 270n^2 - 26436n + 250703, PrimeQ[#] &]

A272159 Numbers k such that abs(8k^2 - 488k + 7243) 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, 64, 65, 66, 67, 71

Offset: 1

Robert Price, Apr 21 2016

nonn

From Robert Israel, Apr 21 2016: (Start)
n such that either n <= 61 or 8n^2 - 488n + 7243 is prime.
The first number not in the sequence is 62. (End)

			4 is in this sequence since 8*4^2 - 488*4 + 7243 = 128-1952+7243 = 5419 is prime.

Robert Israel, 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, A272074, A272075, A272160.

select(n -> isprime(abs(8*n^2 - 488*n + 7243)), [$0..1000]); # Robert Israel, Apr 21 2016

Select[Range[0, 100], PrimeQ[8#^2 - 488# + 7243] &]

lista(nn) = for(n=0, nn, if(isprime(abs(8*n^2-488*n+7243)), print1(n, ", "))); \\ Altug Alkan, Apr 21 2016

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 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, 41, 43, 44, 48, 51, 54, 55, 56, 58, 61, 62, 63, 64, 65, 66, 67, 69, 71, 76, 78, 79, 84, 87, 88, 89, 90, 92

Offset: 1

Robert Price, Apr 23 2016

nonn

40 is the first value not in the sequence.

			4 is in this sequence since 42*4^3 + 270*4^2 - 26436*4 + 250703 = 151967, which is prime.

Robert Price, Table of n, a(n) for n = 1..3092
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050265-A050268, A005846, A007641, A007635, A048988, A256585, A271980, A272074, A272075, A272160, A271144.

Select[Range[0, 100], PrimeQ[42#^3 + 270#^2 - 26436# + 250703] &]

is(n)=isprime(42*n^3+270*n^2-26436*n+250703) \\ Charles R Greathouse IV, Feb 17 2017

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

Original entry on oeis.org

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

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[#] &]

cp's OEIS Frontend

A048988 Primes of the form 4*k^2 + 4*k + 59.

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A272074 Numbers k such that k^4 + 29*k^2 + 101 is prime.

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A272075 Primes of the form k^4 + 29*k^2 + 101.

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A272160 Primes of the form abs(8n^2 - 488n + 7243) in order of increasing nonnegative values of n.

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A271144 Primes of the form 42*k^3 + 270*k^2 - 26436*k + 250703 in order of increasing k.

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A272159 Numbers k such that abs(8*k^2 - 488*k + 7243) is prime.

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Maple

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A271143 Numbers k such that 42*k^3 + 270*k^2 - 26436*k + 250703 is prime.

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Examples

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Mathematica

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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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A048988 Primes of the form 4k^2 + 4k + 59.

A271144 Primes of the form 42k^3 + 270k^2 - 26436*k + 250703 in order of increasing k.

A272159 Numbers k such that abs(8k^2 - 488k + 7243) is prime.

A271143 Numbers k such that 42k^3 + 270k^2 - 26436*k + 250703 is prime.

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.