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A153309 Numbers k such that 3*k + 1 is not prime.

%I A153309 #37 Sep 08 2022 08:45:39
%S A153309 0,1,3,5,7,8,9,11,13,15,16,17,18,19,21,23,25,27,28,29,30,31,33,35,37,
%T A153309 38,39,40,41,43,44,45,47,48,49,51,53,55,56,57,58,59,61,62,63,65,67,68,
%U A153309 69,71,72,73,75,77,78,79,81,82,83,84,85,86,87,88,89,91,93,95
%N A153309 Numbers k such that 3*k + 1 is not prime.
%C A153309 Terms (except 0) can be written as 3xy +- (x + y) for x > 0, y > 0. - _Ron R Spencer_, Aug 01 2016
%C A153309 Apart from a(2) = 1 the sequence comprises those numbers k such that (3*k)!/(3*k + 1) is an integer. - _Peter Bala_, Jan 25 2017
%H A153309 Vincenzo Librandi, <a href="/A153309/b153309.txt">Table of n, a(n) for n = 1..1000</a>
%e A153309 Distribution of the even terms in the following triangular array:
%e A153309                         *;
%e A153309                       *   8;
%e A153309                     *   *  16;
%e A153309                   *   *   *   *;
%e A153309                 *  18   *   *  40;
%e A153309               *   *  30   *   *  56;
%e A153309             *   *   *   *   *   *   *;
%e A153309           *  28   *   *  62   *   *  96;
%e A153309         *   *  44   *   *  82   *   *  120;
%e A153309       *   *   *   *   *   *   *   *   *   *;
%e A153309     *  38   *   *  84   *   *  130  *   *  176;
%e A153309   *   *  58   *   *  108  *   *  158  *   *  208;
%e A153309 etc., where * marks the noninteger values of (4*h*k + 2*k + 2*h)/3 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013
%p A153309 # produces the sequence apart from the term equal to 1
%p A153309 for n from 0 to 100 do
%p A153309 if irem(factorial(3*n), 3*n+1) = 0 then print(n); end if;
%p A153309 end do: # _Peter Bala_, Jan 25 2017
%t A153309 Select[Range[0, 200], !PrimeQ[3*# + 1]&] (* _Vincenzo Librandi_, Jan 12 2013 *)
%o A153309 (Magma) [n: n in [0..150] | not IsPrime(3*n + 1)]; // _Vincenzo Librandi_, Jan 12 2013
%o A153309 (PARI) is(n)=!isprime(3*n+1) \\ _Charles R Greathouse IV_, Aug 01 2016
%Y A153309 Cf. A024892, A014076, A153170, A045751, A095277, A153088, A153329, A153343.
%K A153309 nonn,easy
%O A153309 1,3
%A A153309 _Vincenzo Librandi_, Dec 23 2008
%E A153309 Erroneous comment deleted by _N. J. A. Sloane_, Jun 23 2010
%E A153309 0 added by _Arkadiusz Wesolowski_, Jun 25 2011