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

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

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

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

Programs

Haskell

a002733 = a000196 . (subtract 1) . (* 10) . a207337
-- Reinhard Zumkeller, Apr 06 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 *)

PARI

forstep(n=7,1e3,[6,4],if(isprime(n^2\10+1),print1(n", "))) \\ Charles R Greathouse IV, Mar 11 2012

Formula

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