A002733 Numbers k such that (k^2 + 1)/10 is prime. (Formerly M4342 N1047)

7, 13, 17, 23, 27, 33, 37, 53, 63, 67, 77, 87, 97, 103, 113, 127, 137, 147, 153, 163, 167, 197, 223, 227, 247, 263, 267, 277, 283, 287, 297, 303, 323, 347, 363, 367, 373, 383, 397

Offset

1,1

Comments

Contribution from Wolfdieter Lang, Feb 27 2012: (Start)

The corresponding primes (n^2 + 1)/10 are given in A207337(n).

a(n) is the smallest positive representative of the class of nontrivial solutions of the congruence x^2 == 1 (Modd A207337(n)), if n >= 2. The trivial solution is the class with representative x=1, which also includes -1. For Modd n see a comment on A203571. For n=1: a(1) = 7 == 3 (Modd 5), and 3 is the smallest positive solution > 1.

The unique class of nontrivial solutions of the congruence x^2 == 1 (Modd p), with p an odd prime, exists for any p of the form 4*k+1, given in A002144. Here a subset of these primes is covered, the ones for k = k(n) = (a(n)^2 - 9)/40. These k-values are [1, 4, 7, 13, 18, 27, 34, 70, 99, 112, ...].

(End)

References

L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 25.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

Formula

a(n) = sqrt(10*A207337(n)-1) = sqrt(8*A207339(n)+1), n >= 1. - Wolfdieter Lang, Feb 27 2012

Maple

a := [ ]: for n from 1 to 400 do if (n^2+1 mod 10) = 0 and isprime((n^2+1)/10) then a := [ op(a), n ]; fi; od;

Mathematica

Select[Range[573], PrimeQ[(#^2 + 1)/10] &] (* T. D. Noe, Feb 28 2012 *)

Prog

(PARI) forstep(n=7, 1e3, [6, 4], if(isprime(n^2\10+1), print1(n", "))) \\ Charles R Greathouse IV, Mar 11 2012
(Haskell)
a002733 = a000196 . (subtract 1) . (* 10) . a207337
-- Reinhard Zumkeller, Apr 06 2012

Crossrefs

Cf. A000196, A010051, A002522, A002731, A002732.

Sequence in context: A230039 A226138 A180263 * A108334 A288713 A136083

Adjacent sequences: A002730 A002731 A002732 * A002734 A002735 A002736

Keyword

nonn

Author

N. J. A. Sloane

Status

approved