A090612 -id:A090612 - cp's OEIS frontend

A089001 Numbers n such that 2*n^2 + 1 is prime.

1, 3, 6, 9, 21, 24, 27, 30, 33, 36, 42, 45, 66, 72, 75, 87, 93, 96, 99, 102, 105, 123, 132, 135, 153, 156, 162, 177, 186, 189, 201, 204, 219, 228, 237, 240, 255, 264, 273, 285, 297, 300, 306, 321, 324, 327, 351, 357, 360, 366, 375, 387, 393, 399, 405, 417, 423

Offset: 1

N. J. A. Sloane, Dec 20 2003

All terms except the first one are multiples of 3. - Zak Seidov, Feb 24 2006
And because of this, all the primes except for the first one are congruent to 1 (mod 6). - Robert G. Wilson v, Aug 05 2014
For any n in this sequence, 3*(2*n^2 + 1) has the same nonzero digits as its prime factors in base 2n. - Ely Golden, Dec 12 2016

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

Cf. A089008, A090612, A090698.

[n: n in [1..500] | IsPrime(2*n^2+1)]; // Vincenzo Librandi, Jan 07 2013

Select[Range[500], PrimeQ[2#^2 + 1]&] (* Vincenzo Librandi, Jan 07 2013 *)

is(n)=isprime(2*n^2+1) \\ Charles R Greathouse IV, Feb 17 2017

a(n)=((A090698(n)-1)/2)^(1/2).

Starting with n=2, a(n)=3*A089008(n-1). - Zak Seidov, Feb 24 2006

A090698 Primes of the form 2*n^2+1.

Original entry on oeis.org

3, 19, 73, 163, 883, 1153, 1459, 1801, 2179, 2593, 3529, 4051, 8713, 10369, 11251, 15139, 17299, 18433, 19603, 20809, 22051, 30259, 34849, 36451, 46819, 48673, 52489, 62659, 69193, 71443, 80803, 83233, 95923, 103969, 112339, 115201, 130051

Offset: 1

Kurmang. Aziz. Rashid, Dec 20 2003

A prime p can be expressed as either the sum of two squares or the sum of two squares - 1, p = X^2 + Y^2 or p = X^2 + Y^2 - 1, if and only if p is of the form 2*(m^2)+1 where m is either 1 or a multiple of 3.
Conjecture: 2^(a(n)-1) - 3 is not prime. - Vincenzo Librandi, Feb 04 2013.
Primes in A058331. - Vincenzo Librandi, Apr 10 2015

			19 = 2^2 + 4^2 - 1 = 2*(3^2)+1
73 = 5^2 + 7^2 - 1 = 2*(6^2)+1
163= 8^2 + 10^2 -1 = 2*(9^2)+1
883= 10^2+ 28^2 -1 = 2*(21^2)+1

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A058331, A089001, A089008, A090612.

[a: n in [0..400] | IsPrime(a) where a is 2*n^2+1];// Vincenzo Librandi, Dec 02 2011

Select[Table[2n^2+1,{n,0,900}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

is(n)=isprime(2*n^2+1) \\ Charles R Greathouse IV, Jan 05 2013

a(n)=2*A089001(n)^2+1 = A000040(A090612(n)).

Extended by Ray Chandler, Dec 21 2003

A089008 Numbers k such that 18*k^2 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 10, 11, 12, 14, 15, 22, 24, 25, 29, 31, 32, 33, 34, 35, 41, 44, 45, 51, 52, 54, 59, 62, 63, 67, 68, 73, 76, 79, 80, 85, 88, 91, 95, 99, 100, 102, 107, 108, 109, 117, 119, 120, 122, 125, 129, 131, 133, 135, 139, 141, 142, 143, 147, 150, 152, 154, 156

Offset: 1

N. J. A. Sloane, Dec 20 2003

nonn

There are 8 consecutive terms at n=13537 and n=105819293 for n < 10^9. - Jean C. Lambry, Oct 19 2015

Since 18*k^2 + 1 is divisible by 17 for k == 4, 13 (mod 17), the maximum possible number of consecutive terms is 8, in which case the first term must be congruent to 5 modulo 17 and 7 or 8 modulo 11. - Jianing Song, Nov 14 2021

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A089001, A090612, A090698.

Select[Range[200],PrimeQ[18#^2+1]&]  (* Harvey P. Dale, Apr 25 2011 *)

for(n=0, 1e3, if(isprime(k=(18*n^2 + 1)), print1(n", "))) \\ Altug Alkan, Oct 19 2015

a(n) = A089001(n+1)/3.

A115272 Primes p such that p + 2, 18p^2 + 1, and 18(p+2)^2 + 1 are all primes.

Original entry on oeis.org

29, 107, 431, 1487, 1607, 2141, 5501, 10139, 10271, 17579, 22481, 23057, 27479, 32369, 36341, 36929, 38447, 55931, 57527, 69827, 75539, 78539, 79691, 81047, 81971, 84179, 86027, 89561, 93761, 102059, 112571, 113147, 118799, 119687

Offset: 1

Zak Seidov, Jan 19 2006

nonn

			a(1)=29 because 31, 18*29^2 + 1 = 15139, and 18*31^2 + 1 = 17299 are all primes.

Cf. A089001 (Numbers n such that 2*n^2 + 1 is prime),
A090612 (Numbers k such that the k-th prime is of the form 2*k^2+1),
A090698 (Primes of the form 2*n^2+1),
A113541 (Numbers n such that 18*n^2+1 is a multiple of 19).

[p: p in PrimesUpTo(200000)| IsPrime(p+2) and IsPrime(18*p^2+1) and IsPrime(18*(p+2)^2+1)] // Vincenzo Librandi, Nov 13 2010

More terms from Vincenzo Librandi, Mar 27 2010

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A089001 Numbers n such that 2*n^2 + 1 is prime.

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A090698 Primes of the form 2*n^2+1.

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A089008 Numbers k such that 18*k^2 + 1 is prime.

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Mathematica

PARI

Formula

A115272 Primes p such that p + 2, 18*p^2 + 1, and 18*(p+2)^2 + 1 are all primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Extensions

A115272 Primes p such that p + 2, 18p^2 + 1, and 18(p+2)^2 + 1 are all primes.