A028870 -id:A028870 - cp's OEIS frontend

A028871 Primes of the form k^2 - 2.

2, 7, 23, 47, 79, 167, 223, 359, 439, 727, 839, 1087, 1223, 1367, 1847, 2207, 2399, 3023, 3719, 3967, 4759, 5039, 5623, 5927, 7919, 8647, 10607, 11447, 13687, 14159, 14639, 16127, 17159, 18223, 19319, 21023, 24023, 25919, 28559, 29927

Offset: 1

Patrick De Geest

Except for the initial term, primes equal to the product of two consecutive even numbers minus 1. - Giovanni Teofilatto, Sep 24 2004
With exception of the first term 2, primes p such that continued fraction of (1+sqrt(p))/2 have period 4. - Artur Jasinski, Feb 03 2010
Subsequence of A094786. First primes q that are in A094786 but not here are 241, 3373, 6857, 19681, 29789, 50651, 300761, 371291, ...; q+2 are perfect powers m^k with odd k>1. - Zak Seidov, Dec 09 2014

D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 31.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
P. De Geest, Palindromic Quasipronics of the form n(n+x)
Eric Weisstein's World of Mathematics, Near-Square Prime
Wikipedia, Bunyakovsky conjecture

Cf. A028870, A089623, A010051, A094786; subsequence of A008865.

a028871 n = a028871_list !! (n-1)
a028871_list = filter ((== 1) . a010051') a008865_list
-- Reinhard Zumkeller, May 06 2013

[p: p in PrimesUpTo(100000)| IsSquare(p+2)]; // Vincenzo Librandi, Jun 19 2014

select(isprime, [2,seq((2*n+1)^2-2, n=1..1000)]); # Robert Israel, Dec 09 2014

lst={};Do[s=n^2;If[PrimeQ[p=s-2], AppendTo[lst, p]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 26 2008 *)
aa = {}; Do[If[4 == Length[ContinuedFraction[(1 + Sqrt[Prime[m]])/2][[2]]], AppendTo[aa, Prime[m]]], {m, 1, 1000}]; aa (* Artur Jasinski, Feb 03 2010 *)
Select[Table[n^2 - 2, {n, 400}], PrimeQ] (* Vincenzo Librandi, Jun 19 2014 *)

list(lim)=select(n->isprime(n),vector(sqrtint(floor(lim)+2),k,k^2-2)) \\ Charles R Greathouse IV, Jul 25 2011

a(n) = A028870(n)^2 -2. - R. J. Mathar, Dec 12 2023

A066049 Numbers n such that 2*n^2 - 1 is a prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 17, 18, 21, 22, 24, 25, 28, 34, 36, 38, 39, 41, 42, 43, 45, 46, 49, 50, 52, 56, 59, 62, 63, 64, 69, 73, 76, 80, 81, 85, 87, 91, 92, 95, 98, 102, 108, 109, 112, 113, 115, 118, 125, 126, 127, 132, 134, 137, 140, 141, 143, 153, 154, 155

Offset: 1

N. J. A. Sloane, Jan 09 2002

nonn

It is conjectured that this sequence is infinite.

A066436 gives resulting primes p such that (p+1)/2 is square. - Ray Chandler

D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 31.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A028870, A066436, A090697, A091176, A110558.

Select[Range[200],PrimeQ[2#^2-1]&] (* Harvey P. Dale, Jun 14 2011 *)

{ n=0; for (m=1, 10^9, if (isprime(2*m^2 - 1), write("b066049.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 08 2009

a(n) = A090697(n)/2 = A110558(n)/4. - Ray Chandler, Sep 15 2005

a(n) = A160697(n+1). - Reinhard Zumkeller, May 24 2009

Extended by Ray Chandler, Sep 15 2005

A038599 Numbers k such that k^3 - 2 is prime.

Original entry on oeis.org

9, 15, 19, 27, 31, 37, 67, 91, 99, 109, 121, 129, 135, 145, 151, 165, 187, 189, 201, 207, 211, 217, 241, 259, 265, 267, 277, 279, 289, 319, 355, 357, 367, 369, 387, 391, 411, 417, 427, 435, 439, 445, 457, 459, 477, 489, 511, 525, 549, 555, 561, 615, 619, 621

Offset: 1

Jeff Burch

nonn

			15^3 - 2 = 3373 is prime, so 15 is in the sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A038600, A028870, A028873.

a[n_]:=n^x-y;lst={};x=3;y=2;Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
Select[Range@ 640, PrimeQ[#^3 - 2] &] (* Michael De Vlieger, May 10 2016 *)

is(n)=isprime(n^3-2) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A038600(n)+2)^(1/3). - Zak Seidov, May 10 2016

Missed term, 207, and more terms added by Zak Seidov, Mar 14 2009

A108701 Values of n such that n^2-2 and n^2+2 are both prime.

Original entry on oeis.org

3, 9, 15, 21, 33, 117, 237, 273, 303, 309, 387, 429, 441, 447, 513, 561, 573, 609, 807, 897, 1035, 1071, 1113, 1143, 1233, 1239, 1311, 1563, 1611, 1617, 1737, 1749, 1827, 1839, 1953, 2133, 2211, 2283, 2589, 2715, 2721, 2955, 3081, 3093, 3453, 3549, 3555, 3621, 3723, 3807

Offset: 1

John L. Drost, Jun 19 2005

Since x^2 + 2 is divisible by 3 unless x is divisible by 3, all elements are 3 mod 6.

Intersection of A067201 and A028870. - Robert Israel, Sep 11 2014

			21 is on the list since 21^2 - 2 = 439 and 21^2 + 2 = 443 are primes.

David Wells, Prime Numbers, John Wiley and Sons, 2005, p. 219 (article:'Siamese primes')

Nathaniel Johnston, Table of n, a(n) for n = 1..8750
Raymond A. Beauregard and E. R. Suryanarayan, Square-plus-two primes, Mathematical Gazette 85(502) 90-1.

Subsequence of A016945.

Cf. A016945, A028870, A028873, A038599, A067201, A153974.

[n: n in [3..3600 by 6] | IsPrime(n^2-2) and IsPrime(n^2+2)];  // Bruno Berselli, Apr 15 2011

select(n -> isprime(n^2-2) and isprime(n^2+2), [seq(6*i+3,i=0..1000)]); # Robert Israel, Sep 11 2014

Select[Range[5000], PrimeQ[#^2 - 2] && PrimeQ[#^2 + 2] &] (* Alonso del Arte, Sep 11 2014 *)

is(n)=isprime(n^2-2)&&isprime(n^2+2) \\ Charles R Greathouse IV, Jul 02 2013

Terms corrected by Charles R Greathouse IV, Sep 11 2014

A154831 Numbers n such that n^4-2 is prime.

Original entry on oeis.org

3, 7, 11, 13, 21, 29, 39, 41, 43, 49, 53, 59, 73, 83, 85, 87, 95, 99, 101, 119, 129, 141, 143, 175, 181, 185, 189, 207, 217, 239, 241, 277, 279, 293, 311, 315, 323, 339, 343, 363, 367, 371, 375, 381, 389, 409, 421, 433, 435, 451, 473, 483, 497, 503, 507, 515

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 15 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A028870, A038599.

[n: n in [1..500] | IsPrime(n^4-2)]; // Vincenzo Librandi, Nov 26 2010

lst={};Do[p=n^4-2;If[PrimeQ[p],AppendTo[lst,n]],{n,0,7!}];lst
Select[Range[600],PrimeQ[#^4-2]&] (* Harvey P. Dale, May 20 2012 *)

is(n)=isprime(n^4-2) \\ Charles R Greathouse IV, Jul 02 2013

A153974 Numbers n such that n^3 - 3 is prime.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 26, 34, 38, 40, 74, 80, 106, 110, 116, 124, 136, 158, 178, 184, 190, 206, 224, 230, 238, 256, 274, 280, 316, 320, 338, 340, 386, 410, 428, 446, 458, 464, 470, 484, 496, 530, 544, 550, 556, 590, 626, 634, 644, 646, 674, 710, 718, 728, 730

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

2^3 - 3 = 5 is prime, 4^3 - 3 = 61 is prime, ...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A028870, A028873, A038599, A108701, A153975.

[n: n in [2..500] | IsPrime(n^3-3)]; // Vincenzo Librandi, Nov 26 2010

a[n_]:=n^x-y;lst={};x=3;y=3;Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,0,6!}];lst
Select[Range[2, 1000], PrimeQ[#^3 - 3] &] (* G. C. Greubel, Sep 01 2016 *)

is(n)=isprime(n^3-3) \\ Charles R Greathouse IV, Feb 17 2017

First two terms 0,1, removed by Zak Seidov, Mar 14 2009

A154832 Primes p such that p^4-2 is also prime.

Original entry on oeis.org

3, 7, 11, 13, 29, 41, 43, 53, 59, 73, 83, 101, 181, 239, 241, 277, 293, 311, 367, 389, 409, 421, 433, 503, 587, 617, 647, 683, 757, 773, 811, 823, 839, 881, 907, 953, 1019, 1093, 1117, 1187, 1249, 1361, 1471, 1481, 1543, 1553, 1571, 1637, 1667, 1747, 1789, 1847

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 15 2009

nonn

Primes in A154831.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A028870, A038599, A154831

lst={};Do[p=n^4-2;If[PrimeQ[p],If[PrimeQ[n],AppendTo[lst,n]]],{n,0,7!}];lst
Select[Prime[Range[300]],PrimeQ[#^4-2]&] (* Harvey P. Dale, Nov 24 2018 *)

A239414 Numbers k such that k^6 - 6 is prime.

Original entry on oeis.org

5, 7, 17, 37, 113, 137, 157, 173, 175, 203, 223, 227, 295, 337, 395, 407, 475, 487, 503, 535, 605, 617, 707, 743, 797, 833, 857, 863, 865, 877, 905, 943, 947, 965, 973, 995, 1037, 1043, 1057, 1103, 1217, 1243, 1247, 1277, 1295, 1337, 1357, 1363, 1375, 1403

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all the numbers in this sequence are odd.

			5^6 - 6 = 15619 is prime. Thus, 5 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028870, A109836, A153974.

Select[Range[3,1501,2],PrimeQ[#^6-6]&] (* Harvey P. Dale, Jul 24 2016 *)

is(n)=isprime(n^6-6) \\ Charles R Greathouse IV, Feb 17 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**6-6)}

A154833 Numbers n such that n^5-2 is prime.

Original entry on oeis.org

3, 13, 31, 63, 93, 139, 181, 211, 229, 265, 271, 303, 325, 339, 343, 345, 411, 441, 519, 523, 531, 549, 555, 573, 619, 663, 675, 681, 693, 741, 751, 805, 819, 835, 853, 861, 945, 951, 969, 975, 993, 1063, 1071, 1095, 1119, 1143, 1275, 1281, 1305, 1329

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 15 2009

3^5-2=241 prime, 13^5-2=371291 prime,...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A028870, A038599, A154831, A154832.

[n: n in [1..500] | IsPrime(n^5-2)]; // Vincenzo Librandi, Nov 26 2010

lst={};Do[p=n^5-2;If[PrimeQ[p],AppendTo[lst,n]],{n,0,7!}];lst
Select[Range[2 10^3], PrimeQ[#^5 - 2] &] (* Vincenzo Librandi, Mar 20 2014 *)

is(n)=isprime(n^5-2) \\ Charles R Greathouse IV, Feb 17 2017

A154834 Primes p such that p^5 - 2 is also prime.

Original entry on oeis.org

3, 13, 31, 139, 181, 211, 229, 271, 523, 619, 751, 853, 1063, 1483, 1699, 2791, 3361, 3463, 3541, 3769, 4051, 4201, 4801, 4861, 4903, 5521, 5689, 5701, 6163, 6211, 6763, 6823, 6949, 7621, 8059, 8269, 8389, 8419, 8563, 8689, 8713, 9001, 9103, 9319, 10303

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 15 2009

nonn

Primes in A154833.

			3^5 - 2 = 241 is prime,
13^5 - 2 = 371291 is prime, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028870, A038599, A154831, A154832, A154833.

lst={};Do[p=n^5-2;If[PrimeQ[p],If[PrimeQ[n],AppendTo[lst,n]]],{n,0,7!}];lst
Select[Prime[Range[1300]],PrimeQ[#^5-2]&] (* Harvey P. Dale, Feb 09 2019 *)

cp's OEIS Frontend

A028871 Primes of the form k^2 - 2.

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A066049 Numbers n such that 2*n^2 - 1 is a prime.

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A038599 Numbers k such that k^3 - 2 is prime.

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A108701 Values of n such that n^2-2 and n^2+2 are both prime.

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A154831 Numbers n such that n^4-2 is prime.

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A153974 Numbers n such that n^3 - 3 is prime.

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A154832 Primes p such that p^4-2 is also prime.

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