A090697 -id:A090697 - cp's OEIS frontend

A066436 Primes of the form 2*n^2 - 1.

7, 17, 31, 71, 97, 127, 199, 241, 337, 449, 577, 647, 881, 967, 1151, 1249, 1567, 2311, 2591, 2887, 3041, 3361, 3527, 3697, 4049, 4231, 4801, 4999, 5407, 6271, 6961, 7687, 7937, 8191, 9521, 10657, 11551, 12799, 13121, 14449, 15137, 16561

Offset: 1

N. J. A. Sloane, Jan 09 2002

nonn

It is conjectured that this sequence is infinite.
Also primes p such that 8p + 8 is a square. - Cino Hilliard, Dec 18 2003
Also primes p such that 2p+2 is square; also primes p such that (p+1)/2 is square. - Ray Chandler, Sep 15 2005
Arithmetic numbers which are squares, A003601(p)=A000290(k), p prime, k integer. sigma_1(p)/sigma_0(p)=k^2; p prime, k integer. - Ctibor O. Zizka, Jul 14 2008

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

Harry J. Smith, Table of n, a(n) for n = 1..1000
Project Euler, Problem 216: Investigating the primality of numbers of the form 2n^2-1.

See A066049 for the values of n, see A091176 for prime index.

Cf. A090697, A110558, A003601, A000290.

[ p: n in [1..100] | IsPrime(p) where p is 2*n^2-1 ]; // Klaus Brockhaus, Dec 29 2008

Select[2*Range[200]^2-1,PrimeQ] (* Harvey P. Dale, Aug 29 2016 *)

{ n=0; for (m=1, 10^9, p=2*m^2 - 1; if (isprime(p), write("b066436.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 14 2010

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

A091176 Numbers n such that prime(n) is of the form 2*k^2 - 1.

Original entry on oeis.org

4, 7, 11, 20, 25, 31, 46, 53, 68, 87, 106, 118, 152, 163, 190, 204, 247, 344, 377, 418, 436, 474, 492, 516, 558, 580, 647, 669, 713, 816, 894, 975, 1003, 1028, 1179, 1300, 1392, 1526, 1561, 1695, 1768, 1917, 1952, 2069, 2177, 2343, 2601, 2643, 2769, 2812

Offset: 1

Ray Chandler, Dec 25 2003

nonn

A066436 indexed by A000040.

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

Cf. A066049, A066436, A090697, A110558.

Flatten[Position[Prime[Range[3000]],?(IntegerQ[Surd[(#+1)/2,2]]&)]] (* _Harvey P. Dale, Jan 23 2013 *)

A000040(n) = A066436(n).

A110558 Numbers n such that (n^2-8)/8 is prime.

Original entry on oeis.org

8, 12, 16, 24, 28, 32, 40, 44, 52, 60, 68, 72, 84, 88, 96, 100, 112, 136, 144, 152, 156, 164, 168, 172, 180, 184, 196, 200, 208, 224, 236, 248, 252, 256, 276, 292, 304, 320, 324, 340, 348, 364, 368, 380, 392, 408, 432, 436, 448, 452, 460, 472, 500, 504, 508

Offset: 1

Pierre CAMI, Sep 12 2005

nonn

These numbers need to be of the form 4*j then (16*j^2-8)/8 = 2*j^2-1.

A066436 gives resulting primes p such that 8p+8 is square. - Ray Chandler, Sep 15 2005

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A066049, A066436, A090697, A091176.

Select[Range[510],PrimeQ[(#^2-8)/8]&] (* Harvey P. Dale, Feb 03 2011 *)

is(n)=isprime((n^2-8)/8) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = 2*A090697(n) = 4*A066049(n). - Ray Chandler, Sep 15 2005

Extended by Ray Chandler, Sep 15 2005

cp's OEIS Frontend

A066436 Primes of the form 2*n^2 - 1.

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

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A091176 Numbers n such that prime(n) is of the form 2*k^2 - 1.

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A110558 Numbers n such that (n^2-8)/8 is prime.

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