A007641 -id:A007641 - cp's OEIS frontend

A272410 Primes of the form abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) in order of increasing nonnegative n.

213589, 171329, 135089, 104323, 78509, 57149, 39769, 25919, 15173, 7129, 1409, 2341, 4451, 5227, 4951, 3881, 2251, 271, 1873, 4019, 6029, 7789, 9209, 10223, 10789, 10889, 10529, 9739, 8573, 7109, 5449, 3719, 2069, 673, 271, 541, 109, 1949, 5273, 10399, 17669

Offset: 1

Robert Price, Apr 30 2016

nonn

			78509 is in this sequence since abs(4^4 - 97*4^3 + 3294*4^2 - 45458*4 + 213589) = abs(256-6208+52704-181832+213589) = 78509 is prime.

Robert Price, Table of n, a(n) for n = 1..2676
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.

n = Range[0, 100]; Select[n^4 - 97n^3 + 3294n^2 - 45458n + 213589, PrimeQ[#] &]

A272443 Nonnegative numbers n such that abs(n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397) 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, 50, 51, 53, 57, 58, 59, 64, 67, 70, 75, 79, 80, 81, 89, 91, 92, 93, 96, 99

Offset: 1

Robert Price, Apr 29 2016

nonn

47 is the smallest number not in this sequence.

			4 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, A272160, A271144, A272285, A272401, A272438, A272444.

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

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

A267252 Primes of the form abs(103n^2 - 4707n + 50383) in order of increasing nonnegative n.

Original entry on oeis.org

50383, 45779, 41381, 37189, 33203, 29423, 25849, 22481, 19319, 16363, 13613, 11069, 8731, 6599, 4673, 2953, 1439, 131, 971, 1867, 2557, 3041, 3319, 3391, 3257, 2917, 2371, 1619, 661, 503, 1873, 3449, 5231, 7219, 9413, 11813, 14419, 17231, 20249, 23473, 26903

Offset: 1

Robert Price, Apr 28 2016

This polynomial is a transformed version of the polynomial P(x) = 103*x^2 + 31*x - 3391 whose absolute value gives 43 distinct primes for -23 <= x <= 19, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

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

Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004.

Robert Price, Table of n, a(n) for n = 1..3885
François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.
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, A267069, A330363.

n = Range[0, 100]; Abs @ Select[103n^2 - 4707n + 50383 , PrimeQ[#] &]

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

Title corrected by Hugo Pfoertner, Dec 13 2019

A268200 Nonnegative numbers n such that abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) 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, 51, 52, 53, 54, 62, 65, 67, 70, 72, 73, 74, 75, 84, 85, 86, 90, 92

Offset: 1

Robert Price, Apr 30 2016

nonn

50 is the smallest number not in this sequence.

			4 is in this sequence since abs(4^4 - 97*4^3 + 3294*4^2 - 45458*4 + 213589) = abs(256-6208+52704-181832+213589) = 78509 is prime.

Robert Price, Table of n, a(n) for n = 1..2676
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, A272444, A272410.

Select[Range[0, 100], PrimeQ[#^4 - 97#^3 + 3294#^2 - 45458# + 213589] &]

is(n)=isprime(abs(n^4-97*n^3+3294*n^2-45458*n+213589)) \\ Charles R Greathouse IV, Feb 20 2017

A271348 Primes p such that p + 2*k^2 is prime for at least 10 consecutive values of k starting from k=1.

Original entry on oeis.org

11, 29, 438926021, 1210400879, 7446335849, 31757068151, 33090566651, 33164857769, 40137398219, 45133754591, 46642404071, 100444384301, 114546675671, 144553207691, 159587584529, 161557039991, 166054101539, 210447830009, 227625400031, 236241327599, 254850262949, 272259344081

Offset: 1

Waldemar Puszkarz, Apr 04 2016

nonn

Number 10 was chosen as a threshold as the smallest two digit number. You can choose other numbers and if they are less than 12, the first terms of sequences analogous to this one will be those in A165234.
There are 20 primes like that among the first 10^10 of them. The second term, 29, generates 28 primes (A007641). Sixteen others, including 11 (A050265), generate only 10 primes, while three produce 11 primes. These three are: 33164857769 (see also A165234), 159587584529, and 236241327599. The first term among the second 10^10 of primes is 254850262949. Then there is 272259344081 (mentioned in A165234) that generates 13 primes.
All these primes end with 1 or 9 and are congruent to 5 mod 6.

			11 is a term because 11+2*k^2 gives rise to 10 primes for 10 consecutive values of k starting from 1 (see A050265).

Eric Weisstein's World of Mathematics, Prime-generating Polynomial

Cf. A000040 (primes), A050265, A007641, A271366, A271818, A271819, A271820 (examples of sequences of primes generated by terms of this sequence), A165234.

lst={}; Do[k=1; While[PrimeQ[Prime[n]+2*k^2], k++]; If[k>10, AppendTo[lst, Prime[n]]], {n, 2, 11*10^9}]; lst
Select[Prime[Range[107669*10^5]],AllTrue[#+{2,8,18,32,50,72,98,128,162,200},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* The program will take a long time to run *) (* Harvey P. Dale, Jan 24 2021 *)

forprime(n=2, 276241327599, k=1; while(isprime(n+2*k^2), k++); (k>10)&&print1(n, ", "))

A271366 Primes of the form 272259344081 + 2*n^2.

Original entry on oeis.org

272259344081, 272259344083, 272259344089, 272259344099, 272259344113, 272259344131, 272259344153, 272259344179, 272259344209, 272259344243, 272259344281, 272259344323, 272259344369, 272259344419, 272259344881, 272259345433, 272259345539, 272259347123, 272259347281, 272259347953

Offset: 1

Waldemar Puszkarz, Apr 05 2016

nonn

The first 14 primes correspond to the values of n from 0 to 13. The first term is a member of A271348 and A165234.

			For n=0, we get 272259344081, which is a prime as determined in A271348.
For n=1, we get 272259344081 + 2*1^2 = 272259344083, which is a prime as determined in A271348.

Eric Weisstein's World of Mathematics, Prime-generating Polynomial

Cf. A000040 (primes), A271348, A165234 (sequences containing the first term), A050265, A007641, A271818, A271819, A271820 (similar sequences whose first term is in A271348).

Select[Table[272259344081+2*n^2, {n, 0, 100}], PrimeQ]

for(n=0, 100, isprime(272259344081+2*n^2) && print1(272259344081+2*n^2, ","))

A271818 Primes of the form 33164857769 + 2*n^2.

Original entry on oeis.org

33164857769, 33164857771, 33164857777, 33164857787, 33164857801, 33164857819, 33164857841, 33164857867, 33164857897, 33164857931, 33164857969, 33164858011, 33164858347, 33164858569, 33164858737, 33164859019, 33164859569, 33164859691, 33164859817, 33164860219, 33164860507, 33164862769, 33164863177, 33164864731, 33164864969, 33164865457, 33164865961, 33164866481, 33164868427, 33164869321

Offset: 1

Waldemar Puszkarz, Apr 14 2016

nonn

The first 12 primes correspond to the values of n from 0 to 11. The first term is a member of A271348 and A165234.

			For n=0, we get 33164857769, which is a prime as determined in A271348.
For n=1, we get 33164857769 + 2*1^2 = 33164857771, which is a prime as determined in A271348.

Eric Weisstein's World of Mathematics, Prime-generating Polynomial

Cf. A000040 (primes), A271348, A165234 (sequences containing the first term), A050265, A007641, A271366, A271819, A271820 (similar sequences whose first term is in A271348).

Select[Table[33164857769+2*n^2, {n, 0, 100}], PrimeQ]

for(n=0, 100, isprime(33164857769+2*n^2) && print1(33164857769+2*n^2, ", "))

A271819 Primes of the form 159587584529 + 2*n^2.

Original entry on oeis.org

159587584529, 159587584531, 159587584537, 159587584547, 159587584561, 159587584579, 159587584601, 159587584627, 159587584657, 159587584691, 159587584729, 159587584771, 159587585107, 159587585329, 159587585681, 159587585881, 159587586097, 159587586451, 159587586707, 159587586979

Offset: 1

Waldemar Puszkarz, Apr 14 2016

nonn

The first 12 primes correspond to the values of n from 0 to 11. The first term is a member of A271348.

			For n=0, we get 159587584529, which is a prime as determined in A271348.
For n=1, we get 159587584529 + 2*1^2 = 159587584531, which is a prime as determined in A271348.

Eric Weisstein's World of Mathematics, Prime-generating Polynomial

Cf. A000040 (primes), A271348 (contains the first term), A050265, A007641, A271366, A271818, A271820 (similar sequences whose first term is in A271348).

Select[Table[159587584529+2*n^2, {n, 0, 100}], PrimeQ]

for(n=0, 100, isprime(159587584529+2*n^2) && print1(159587584529+2*n^2, ", "))

A271820 Primes of the form 236241327599 + 2*n^2.

Original entry on oeis.org

236241327599, 236241327601, 236241327607, 236241327617, 236241327631, 236241327649, 236241327671, 236241327697, 236241327727, 236241327761, 236241327799, 236241327841, 236241328177, 236241328751, 236241330049, 236241331831, 236241332207, 236241332401, 236241333649, 236241334799

Offset: 1

Waldemar Puszkarz, Apr 14 2016

nonn

The first 12 primes correspond to the values of n from 0 to 11. The first term is a member of A271348.

			For n=0, we get 236241327599, which is a prime as determined in A271348.
For n=1, we get 236241327599 + 2*1^2 = 236241327601, which is a prime as determined in A271348.

Eric Weisstein's World of Mathematics, Prime-generating Polynomial

Cf. A000040 (primes), A271348 (contains the first term), A050265, A007641, A271366, A271818, A271819 (similar sequences whose first term is in A271348).

Select[Table[236241327599+2*n^2, {n, 0, 100}], PrimeQ]

for(n=0, 100, isprime(236241327599+2*n^2) && print1(236241327599+2*n^2, ", "))

A272555 Primes of the form abs(1/(36)(n^6 - 126n^5 + 6217n^4 - 153066n^3 + 1987786n^2 - 13055316n + 34747236)) in order of increasing nonnegative n.

Original entry on oeis.org

965201, 653687, 429409, 272563, 166693, 98321, 56597, 32969, 20873, 15443, 13241, 12007, 10429, 7933, 4493, 461, 3583, 6961, 9007, 9157, 7019, 2423, 4549, 13553, 23993, 35051, 45737, 54959, 61613, 64693, 63421, 57397, 46769, 32423, 16193, 1091, 8443, 6271

Offset: 1

Robert Price, May 02 2016

nonn

			166693 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, A272159, A271143, A272284, A272302, A272437, A272443, A268200, A272554.

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

cp's OEIS Frontend

A272410 Primes of the form abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) in order of increasing nonnegative n.

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A272443 Nonnegative numbers n such that abs(n^5 - 99n^4 + 3588n^3 - 56822n^2 + 348272n - 286397) is prime.

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A267252 Primes of the form abs(103*n^2 - 4707*n + 50383) in order of increasing nonnegative n.

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A268200 Nonnegative numbers n such that abs(n^4 - 97n^3 + 3294n^2 - 45458n + 213589) is prime.

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A271348 Primes p such that p + 2*k^2 is prime for at least 10 consecutive values of k starting from k=1.

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A271366 Primes of the form 272259344081 + 2*n^2.

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A271818 Primes of the form 33164857769 + 2*n^2.

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A271819 Primes of the form 159587584529 + 2*n^2.

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A267252 Primes of the form abs(103n^2 - 4707n + 50383) in order of increasing nonnegative n.