A002731 Numbers k such that (k^2 + 1)/2 is prime. (Formerly M2444 N0971)

3, 5, 9, 11, 15, 19, 25, 29, 35, 39, 45, 49, 51, 59, 61, 65, 69, 71, 79, 85, 95, 101, 121, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 195, 199, 201, 205, 209, 219, 221, 231, 245, 261, 271, 275, 279, 289, 299, 309, 315, 321, 325, 329, 335, 345, 349, 371, 375, 379, 391, 399, 405

Offset

1,1

Comments

From Wolfdieter Lang, Feb 24 2012: (Start)

a(n) = sqrt(8*A129307(n)+1) = sqrt(2*A027862(n)-1), n >= 1.

a(n) is the nontrivial solution of the congruence a(n)^2 == 1 (Modd A027862(n)). The trivial one is +1. For Modd n see a comment on A203571. E.g., a(3)^2 = 81 == 1 (Modd 41), see a comment on A027862.

(End)

References

L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 24.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

L. Euler, De numeris primis valde magnis (E283), The Euler Archive.

Formula

a(n) = 2*A027861(n) + 1.

Mathematica

Select[Range[400], PrimeQ[(#^2 + 1)/2] &] (* Alonso del Arte, Feb 24 2012 *)

Prog

(PARI)
forstep(n=1, 10^3, 2, if(isprime((n^2+1)/2), print1(n, ", ")));
/* Joerg Arndt, Sep 02 2012 */
(Magma) [n: n in [3..410] | IsPrime((n^2+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012
(Haskell)
a002731 n = a002731_list !! (n-1)
a002731_list = filter ((== 1) . a010051 . a000982) [1, 3 ..]
-- Reinhard Zumkeller, Jul 13 2014

Crossrefs

Cf. A027861. A027862 gives primes, A091277 gives prime indices.

Cf. A000982, A010051, A116945, A129307, A188546, A188547, A187431.

Sequence in context: A075991 A292918 A216091 * A006046 A161830 A151922

Adjacent sequences: A002728 A002729 A002730 * A002732 A002733 A002734

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved