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%I #52 Sep 08 2022 08:44:48
%S 2,4,6,10,12,14,20,22,24,26,32,34,36,42,46,50,52,54,60,64,66,70,74,76,
%T 80,90,92,94,102,104,110,112,116,122,124,126,132,136,140,144,146,152,
%U 154,162,166,174,180,182,190,192,200,202,204,206,210,214,220,224,230,236,242,244,246
%N Numbers k such that 3*k+1 is prime.
%C Every prime (with the exception of 3) can be expressed as 3*k+1 or 3*k-1. - _César Aguilera_, Apr 13 2013
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A024892/b024892.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = (A002476(n) - 1)/3. See the name.
%F a(n) = 2*A024899(n) = A034936(n) + 1.
%F a(n) = A153183(n) - 1 = A107303(n) - 2.
%t Select[Range[250], PrimeQ[3# + 1] &] (* _Vincenzo Librandi_, Sep 25 2012 *)
%o (Magma) [n: n in [1..1000] | IsPrime(3*n+1)]; // _Vincenzo Librandi_, Nov 20 2010
%o (PARI) is(n)=isprime(3*n+1) \\ _Charles R Greathouse IV_, Feb 17 2017
%Y Cf. A002476 (associated primes), A091178 (gives prime index).
%Y Cf. A153183, A107303.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_
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