A062395 a(n) = 8^n + 1.

2, 9, 65, 513, 4097, 32769, 262145, 2097153, 16777217, 134217729, 1073741825, 8589934593, 68719476737, 549755813889, 4398046511105, 35184372088833, 281474976710657, 2251799813685249, 18014398509481985, 144115188075855873

Offset: 0

Author

Henry Bottomley, Jun 22 2001

Keywords

• easy
• nonn

Comments

Any number of the form b^k+1 is composite for b>2 and k odd since b+1 algebraically divides b^k+1. - Robert G. Wilson v, Aug 25 2002

References

• D. M. Burton, Elementary Number Theory, Allyn and Bacon, Boston, MA, 1976, pp. 51.
• G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..140
• Index entries for linear recurrences with constant coefficients, signature (9,-8).

Crossrefs

Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062396, A062397, A007689, A063376, A063481, A074600 - A074624, A034524, A178248, A228081 for numbers one more than powers.

Programs

Magma

[8^n + 1: n in [0..40] ]; // Vincenzo Librandi, Apr 30 2011

Mathematica

Table[8^n + 1, {n, 0, 20}]
LinearRecurrence[{9,-8},{2,9},20] (* Harvey P. Dale, Jan 24 2019 *)

PARI

for(n=0,22,print(8^n+1))

Formula

a(n) = 8a(n-1)-7 = A001018(n)+1 = 9a(n-1) - 8a(n-2).

G.f.: -(-2+9*x)/(-1+x)/(-1+8*x). - R. J. Mathar, Nov 16 2007

E.g.f.: e^x+e^(8*x). - Mohammad K. Azarian, Jan 02 2009