cp's OEIS Frontend

A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.

%I A161896 #16 Sep 20 2024 06:30:35
%S A161896 5,11,23,29,41,53,83,89,113,131,173,179,191,233,239,251,281,293,359,
%T A161896 419,431,443,491,509,593,641,653,659,683,719,743,761,809,911,953,1013,
%U A161896 1019,1031,1049,1103,1223,1229,1289,1409,1439,1451,1481,1499,1511,1541,1559
%N A161896 Integers n for which k = (9^n - 3 * 3^n - 4n) / (2n * (2n + 1)) is an integer.
%C A161896 Near superset of the Sophie Germain primes (A005384), excluding 2 and 3: 2n + 1 is prime. Nearly all members of this sequence are also prime, but four members less than 10000 are composite: 1541 = 23 * 67, 2465 = 5 * 17 * 29, 3281 = 17 * 193, and 4961 = 11^2 * 41.
%C A161896 The congruence of n modulo 4 is evenly distributed between 1 and 3. n is congruent to 5 (mod 6) for all n less than two billion.
%C A161896 This sequence has roughly twice the density of the sequence (A158034) corresponding to the Diophantine equation
%C A161896 f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)),
%C A161896 and contains most members of that sequence. Those it does not contain are composite and often congruent to 3 (mod 6).
%C A161896 Composite terms appear to predominantly belong to A262051. - _Bill McEachen_, Aug 29 2024
%H A161896 Reinhard Zumkeller, <a href="/A161896/b161896.txt">Table of n, a(n) for n = 1..1000</a>
%o A161896 (Haskell)
%o A161896 a161896 n = a161896_list !! (n-1)
%o A161896 a161896_list = [x | x <- [1..],
%o A161896                     (9^x - 3*3^x - 4*x) `mod` (2*x*(2*x + 1)) == 0]
%o A161896 -- _Reinhard Zumkeller_, Jan 12 2014
%o A161896 (PARI) is(n)=my(m=2*n*(2*n+1),t=Mod(3,m)^n); t^2-3*t==4*n \\ _Charles R Greathouse IV_, Nov 25 2014
%Y A161896 Cf. A161897 A000040, A002515, A005384, A158034, A158035, A158036, A145918, A002943.
%K A161896 easy,nonn
%O A161896 1,1
%A A161896 _Reikku Kulon_, Jun 21 2009