A089008 -id:A089008 - 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

A090612 Numbers k such that the k-th prime is of the form 2*j^2 + 1.

Original entry on oeis.org

2, 8, 21, 38, 153, 191, 232, 279, 327, 378, 493, 559, 1086, 1272, 1360, 1769, 1989, 2111, 2224, 2344, 2471, 3272, 3721, 3863, 4838, 5006, 5359, 6291, 6871, 7077, 7909, 8127, 9245, 9928, 10654, 10889, 12164, 12957, 13764, 14881, 16034, 16343, 16944

Offset: 1

Ray Chandler, Dec 21 2003

nonn

A090698 indexed by A000040.

			From _Jon E. Schoenfield_, Jan 24 2018: (Start)
prime(8) = 19 = 2*3^2 + 1, so 8 is in the sequence.
prime(21) = 73 = 2*6^2 + 1, so 21 is in the sequence.
prime(33) = 137 = 2*68 + 1, and 68 is not a square, so 33 is not in the sequence. (End)

Robert Israel, Table of n, a(n) for n = 1..1525

Cf. A089001, A089008, A090698.

N:= 1000; # to get all entries corresponding to primes <= 2*N^2+1.
R:= select(isprime,[seq(2*k^2+1,k=1..N)]):
A090612:= map(numtheory[pi],R); # Robert Israel, May 09 2014

Select[Range[18000],IntegerQ[Sqrt[(Prime[#]-1)/2]]&] (* Harvey P. Dale, Apr 25 2016 *)

a(n)=k such that A000040(k) = A090698(n) = 2*A089001(n)^2 + 1.

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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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A090612 Numbers k such that the k-th prime is of the form 2*j^2 + 1.

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