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

5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1541, 1559

Offset: 1

Author

Reikku Kulon, Jun 21 2009

Keywords

• easy
• nonn

Comments

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.

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.

This sequence has roughly twice the density of the sequence (A158034) corresponding to the Diophantine equation

f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)),

and contains most members of that sequence. Those it does not contain are composite and often congruent to 3 (mod 6).

Composite terms appear to predominantly belong to A262051. - Bill McEachen, Aug 29 2024

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A161897 A000040, A002515, A005384, A158034, A158035, A158036, A145918, A002943.

Programs

Haskell

a161896 n = a161896_list !! (n-1)
a161896_list = [x | x <- [1..],
                    (9^x - 3*3^x - 4*x) `mod` (2*x*(2*x + 1)) == 0]
-- Reinhard Zumkeller, Jan 12 2014

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