A028870 -id:A028870 - cp's OEIS frontend

A153975 Values of n such that n^2-3 and n^2+3 are both prime.

4, 8, 10, 14, 64, 92, 112, 140, 146, 172, 218, 298, 304, 322, 326, 340, 350, 356, 416, 440, 470, 508, 554, 560, 580, 626, 634, 652, 668, 686, 694, 704, 728, 736, 746, 770, 806, 818, 868, 892, 920, 1054, 1082, 1102, 1130, 1156, 1196, 1256, 1264, 1378, 1418

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Intersection of A028873 and A049422. - Zak Seidov, Oct 12 2014

			4^2 - 3 = 13 and 4^2 + 3 = 19 are both primes, so 4 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A028870, A028873, A038599, A049422, A108701, A153974.

[n: n in [1..1400] | IsPrime(n^2-3) and IsPrime(n^2+3)]; // Vincenzo Librandi, Oct 12 2014

Select[Range[1500], PrimeQ[#^2 - 3] && PrimeQ[#^2 + 3] &] (* Vincenzo Librandi, Oct 12 2014 *)

is(n) = isprime(n^2-3) && isprime(n^2+3); \\ Altug Alkan, Sep 01 2016

Incorrect term 0 removed and Mma edited by Zak Seidov, Oct 12 2014

A154936 Primes in A154935.

Original entry on oeis.org

7, 199, 211, 337, 367, 1231, 1321, 1627, 1741, 2161, 2251, 2551, 3259, 3769, 3877, 3931, 4099, 4591, 4759, 4789, 6829, 7297, 7867, 8221, 8887, 9049, 9181, 9337, 9349, 11959, 12697, 12919, 13411, 13591, 14827, 15187, 15217, 15817, 15877, 15889

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

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

Cf. A028870, A038599, A154831, A154832, A154833, A154834, A154933, A154934, A154935.

lst={}; Do[p=n^7-2; If[PrimeQ[p], If[PrimeQ[n], AppendTo[lst,n]]], {n,0,8!}]; lst

A296507 Numbers m such that m^2 - 13 is a prime.

Original entry on oeis.org

4, 6, 12, 18, 24, 30, 36, 54, 72, 84, 90, 96, 102, 114, 120, 138, 168, 186, 198, 204, 210, 216, 228, 240, 276, 294, 318, 330, 354, 360, 372, 378, 402, 414, 438, 444, 456, 480, 498, 504, 588, 600, 612, 618, 630, 636, 666, 678, 690, 714, 720, 726, 732, 738, 762

Offset: 1

Zak Seidov, Dec 13 2017

nonn

All terms except 4 are divisible by 6. - Robert Israel, Dec 13 2017

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000

Cf. A028870, A028873, A028876, A028882, A090696.

select(n -> isprime(n^2-13), 2*[$2..10^4]); # Robert Israel, Dec 13 2017

Reap[m=4;Do[If[PrimeQ[m^2-13],Sow[m]];m=m+2,{1000}]][[2,1]]
Select[Range[800],PrimeQ[#^2-13]&] (* Harvey P. Dale, Mar 06 2023 *)

isok(n) = isprime(n^2-13); \\ Michel Marcus, Dec 14 2017

A071351 Numbers n such that both n^4 + 2 and n^4 - 2 are prime.

Original entry on oeis.org

3, 21, 87, 99, 129, 141, 279, 627, 657, 777, 783, 795, 1653, 1725, 1833, 1959, 2001, 2043, 3039, 3399, 3609, 3861, 3975, 4257, 4371, 4491, 5403, 5541, 5709, 5985, 7371, 7539, 7869, 7917, 8397, 8445, 8547, 8793, 9051, 9057, 9915, 9933, 11067, 12153

Offset: 1

Labos Elemer, May 21 2002

			n=3: n^4 = 81; {79,83} are primes.

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

Cf. A028870, A038599, A154831, A154832, A154833, A154834, A154933, A154934, A154935, A154936. - Vladimir Joseph Stephan Orlovsky, Jan 17 2009

lst={}; Do[p1=n^4-2; p2=n^4+2; If[PrimeQ[p1]&&PrimeQ[p2],AppendTo[lst,n]],{n,0,8!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 17 2009 *)
Select[Range[730000], AllTrue[#^4 + {2, -2}, PrimeQ] &] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 02 2018 *)

A140454 Number of primes p less than 10^n such that p^2-2 is prime.

Original entry on oeis.org

4, 13, 52, 259, 1595, 10548, 74914, 563533, 4387106

Offset: 1

Jonathan Vos Post, Jun 26 2008

nonn

Korevaar gives these values in Table 1, p. 18, attributing the calculation to Fokko van de Bult. Abstract: For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an explicit positive constant C_{2r}. A heuristic argument indicates that the remainder e_{2r}(x) in this approximation cannot be of lower order than x^beta, where beta is the supremum of the real parts of zeta's zeros. The argument also suggests an approximation for pi_{2r}(x) similar to one of Riemann for pi(x).

			a(1) = 4 because {2, 3, 5, 7} are the 4 primes p less than 10^1 such that p^2-2 are primes, namely {2, 7, 23, 47}.
a(2) = 13 = #{2, 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89}.

Jacob Korevaar, Lower bound for the remainder in the prime-pair conjecture, arXiv:0806.4057

Cf. A000040, A028870.

a(n) = #{p < 10^n in A028870}.

a(9) from Donovan Johnson, Feb 17 2010

A225098 Numbers k such that k^2 - 2 and 2*k^2 - 1 are both prime.

Original entry on oeis.org

2, 3, 7, 13, 15, 21, 43, 49, 63, 69, 127, 155, 183, 211, 231, 237, 259, 265, 273, 293, 301, 323, 335, 391, 435, 441, 447, 489, 505, 573, 595, 671, 713, 715, 743, 757, 797, 811, 951, 959, 973, 979, 987, 993, 1035, 1147, 1197, 1287, 1359, 1393, 1415, 1429, 1443, 1491, 1525, 1597, 1617, 1653

Offset: 1

Gerasimov Sergey, Apr 27 2013

nonn

Primes in the sequence: 2, 3, 7, 13, 43, 127, 211, 293, 743, 757, 797, 811, 1429,...

			2^2 - 2 = 2 is prime and 2*2^2 - 1 = 7 is prime, so a(1) = 2.

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

Intersection of A028870 and A066049.

Select[Range[1653], PrimeQ[#^2 - 2] && PrimeQ[2*#^2 - 1] &] (* T. D. Noe, May 10 2013 *)

Corrected by R. J. Mathar, May 05 2013

A146980 Nonsquarefree numbers such that n-1 is prime and n+1 is square.

Original entry on oeis.org

8, 24, 48, 80, 168, 224, 360, 440, 728, 840, 1088, 1224, 1368, 1848, 2208, 2400, 3024, 3720, 3968, 4760, 5040, 5624, 5928, 7920, 8648, 10608, 11448, 13688, 14160, 14640, 16128, 17160, 18224, 19320, 21024, 24024, 25920, 28560, 29928, 31328, 33488

Offset: 1

Giovanni Teofilatto, Nov 04 2008

nonn

Also numbers n > 3 such that n-1 is prime and n+1 is square.

Sequence gives values x of fundamental solution (x,y) to Pellian x^2 - D*y^2 = 1, with D = n-1 = A049002, corresponding values y being sqrt(n+1) = A028870. (Substituting back into the Pellian we indeed have n^2 - (n-1)(n+1) = 1.) - Lekraj Beedassy, Feb 23 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A013929, A049002.

[ n: n in [1..35000] | not IsSquarefree(n) and IsPrime(n-1) and IsSquare(n+1) ]; // Klaus Brockhaus, Nov 05 2008

Select[Range[35000], !SquareFreeQ[#] && PrimeQ[#-1] && IntegerQ[Sqrt[#+1] ] &] (* G. C. Greubel, Feb 22 2019 *)
Mean/@SequencePosition[Table[Which[PrimeQ[n],1,IntegerQ[Sqrt[ n]],3,!SquareFreeQ[ n],2,True,0],{n,33500}],{1,2,3}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 03 2020 *)

list(lim)=my(v=List()); forstep(k=3,sqrtint(lim\1+1),2, if(isprime(k^2-2), listput(v,k^2-1))); Vec(v) \\ Charles R Greathouse IV, Jun 13 2017

[n for n in (1..35000) if not is_squarefree(n) and is_prime(n-1) and is_square(n+1)] # G. C. Greubel, Feb 22 2019

Extended beyond a(6) by Klaus Brockhaus, Nov 05 2008

A154938 Numbers k such that k^6 - 2 and k^6 + 2 are both primes.

Original entry on oeis.org

195, 213, 231, 657, 1563, 1749, 2967, 3597, 3627, 4263, 4887, 6867, 6993, 7257, 7725, 9045, 9201, 9717, 11595, 12579, 13029, 14145, 14259, 14367, 15837, 16131, 16581, 17259, 19905, 19917, 21081, 21711, 23127, 24435, 24921, 28299, 28707

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

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

Intersection of A154933 and A216978.

Cf. A028870, A038599, A154831, A154832, A154833, A154834, A154934, A154935, A154936, A071351.

[n: n in [1..500] | IsPrime(n^6-2) and IsPrime(n^6+2)]; // Vincenzo Librandi, Nov 26 2010

lst={};Do[p1=n^6-2;p2=n^6+2;If[PrimeQ[p1]&&PrimeQ[p2],AppendTo[lst,n]],{n,0,9!}];lst
Select[Range[30000],AllTrue[#^6+{2,-2},PrimeQ]&] (* Harvey P. Dale, Jun 21 2025 *)

A216945 Numbers k such that k-2, k^2-2, k^3-2, k^4-2 and k^5-2 are all prime.

Original entry on oeis.org

15331, 289311, 487899, 798385, 1685775, 1790991, 1885261, 1920619, 1967925, 2304805, 2479735, 3049201, 3114439, 3175039, 3692065, 4095531, 4653649, 5606349, 5708235, 6113745, 6143235, 6697425, 7028035, 7461601, 8671585, 8997121, 9260131, 10084915, 10239529

Offset: 1

Michel Lagneau, Sep 20 2012

nonn

k^6-2 is also prime for k = 1685775, 4095531, 4653649, 5606349, 13219339, 13326069, 18439561, ...

Sequence is infinite under Schinzel's Hypothesis H. a(n) >> n log^5 n. - Charles R Greathouse IV, Sep 20 2012

Cf. A052147, A028870, A038599, A154831, A154833, A154935.

Select[Range[20000000], And@@PrimeQ/@(Table[n^i-2, {i, 1, 5}]/.n->#)&]
Select[Prime[Range[680000]]+2,AllTrue[#^Range[2,5]-2,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 11 2020 *)

Sequence is A052147 intersection A028870 intersection A038599 intersection A154831 intersection A154833.

A239474 Smallest k >= 1 such that k^n-n is prime. a(n) = 0 if no such k exists.

Original entry on oeis.org

3, 2, 2, 0, 4, 5, 60, 3, 2, 21, 28, 5, 2, 199, 28, 0, 234, 11, 2, 3, 2, 159, 10, 31, 68, 145, 0, 69, 186, 163, 32, 253, 26, 261, 4, 0, 8, 11, 62, 3, 22, 43, 6, 7, 8, 945, 76, 7, 116, 129, 382, 93, 330, 361, 2, 555, 224, 1359, 78, 29, 62, 39, 110, 0, 1032, 37, 462, 29

Offset: 1

Derek Orr, Mar 20 2014

nonn

If n is of the form (pk)^p for some k and some prime p, then a(n) = 0 (See A097764).

			1^1-1 = 0 is not prime. 2^1-1 = 1 is not prime. 3^1-1 = 2 is prime. Thus, a(1) = 3.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418.

import sympy
from sympy import isprime
def TwoMin(x):
  for k in range(1,5000):
    if isprime(k**x-x):
      return k
x = 1
while x < 100:
  print(TwoMin(x))
  x += 1

a(A097764(n)) = 0 for all n.

cp's OEIS Frontend

A153975 Values of n such that n^2-3 and n^2+3 are both prime.

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Magma

Mathematica

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Extensions

A154936 Primes in A154935.

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Mathematica

A296507 Numbers m such that m^2 - 13 is a prime.

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Maple

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A071351 Numbers n such that both n^4 + 2 and n^4 - 2 are prime.

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Mathematica

A140454 Number of primes p less than 10^n such that p^2-2 is prime.

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Extensions

A225098 Numbers k such that k^2 - 2 and 2*k^2 - 1 are both prime.

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Mathematica

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A146980 Nonsquarefree numbers such that n-1 is prime and n+1 is square.

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Magma

Mathematica

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Sage

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A154938 Numbers k such that k^6 - 2 and k^6 + 2 are both primes.

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