A024892 Numbers k such that 3*k+1 is prime.

2, 4, 6, 10, 12, 14, 20, 22, 24, 26, 32, 34, 36, 42, 46, 50, 52, 54, 60, 64, 66, 70, 74, 76, 80, 90, 92, 94, 102, 104, 110, 112, 116, 122, 124, 126, 132, 136, 140, 144, 146, 152, 154, 162, 166, 174, 180, 182, 190, 192, 200, 202, 204, 206, 210, 214, 220, 224, 230, 236, 242, 244, 246

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Every prime (with the exception of 3) can be expressed as 3*k+1 or 3*k-1. - César Aguilera, Apr 13 2013

The associated prime A002476(n) has a unique representation as x^2 + x*y - 2*y^2 = (x + 2*y)*(x-y) with positive integers, namely (x(n), y(n)) = (a(n) + 1, a(n)). See the N. J. A. Sloane, May 31 2014, comment on A002476. - Wolfdieter Lang, Feb 09 2016

For all elements of this sequence there are no (x,y) positive integers such that a(n) = 3*x*y + x + y or a(n) = 3*x*y - x - y. - Pedro Caceres, Jan 28 2021

Links

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

Crossrefs

Cf. A002476 (associated primes), A091178 (gives prime index).

Cf. A153183, A107303.

Programs

Magma

[n: n in [1..1000] | IsPrime(3*n+1)]; // Vincenzo Librandi, Nov 20 2010

Mathematica

Select[Range[250], PrimeQ[3# + 1] &] (* Vincenzo Librandi, Sep 25 2012 *)

PARI

is(n)=isprime(3*n+1) \\ Charles R Greathouse IV, Feb 17 2017

Formula

a(n) = (A002476(n) - 1)/3. See the name.

a(n) = 2*A024899(n) = A034936(n) + 1.

a(n) = A153183(n) - 1 = A107303(n) - 2.