A093428 Prime numbers which are successors of a power of a Fibonacci number.

5, 17, 257, 65537, 1336337, 19170731299728100000001, 285347346718226949041792907369577650673693754163660005676181161059099319730177, 29585383599687066848440635756425168157198788892517565295922752892368299949134315777

Offset

1,1

Comments

Let a(n) = Fibonacci(x)^y+1, then there exists some a,b > 0, such that x = 3*a and y = 2^b. For the example a(5) = 1336337: x = 9, y = 4, a = 3 and b = 2.

Last digit seems to be usually 7, except for a(1) and a(6). - Alexander Adamchuk, Aug 09 2006

Links

Table of n, a(n) for n=1..8.

U. Dudley, B. Tucker, Greatest Common Divisors in Altered Fibonacci Sequences, Fibonacci Quarterly 1971, pages 89-91.

Letter from Toby Gee in Mathematical Spectrum, Fibonacci numbers, vol. 29 (1996/1997), page 66.

V. E. Hoggatt Jr. and M. Bicknell-Johnson, Composites and Primes Among Powers of Fibonacci Numbers increased or decreased by one, Fibonacci Quarterly vol. 15 (1977), page 2.

Ron Knott, The Mathematical Magic of the Fibonacci Numbers.

Example

a(5) = 1336337 because 1336337 is prime, and 1336337-1 = 1336336 = 34^4+1 = Fibonacci(9)^4+1.
a(6) = Fibonacci(15)^8 + 1, a(7) = Fibonacci(48)^8 + 1, a(8) = Fibonacci(51)^8 + 1, a(9) = Fibonacci(63)^8 + 1, a(10) = Fibonacci(21)^32 + 1, a(11) = Fibonacci(198)^4 + 1, a(12) = Fibonacci(204)^8 + 1, a(13) = Fibonacci(366)^8 + 1. - Alexander Adamchuk, Aug 09 2006

Mathematica

Select[Take[Intersection[Flatten[Table[Fibonacci[3n]^(2^m)+1, {n, 1, 300}, {m, 1, 7}]]], {1, 400}], PrimeQ] (* Alexander Adamchuk, Aug 09 2006 *)

Crossrefs

Cf. A005478, A001605, A000045.

Sequence in context: A273999 A222008 A274000 * A274002 A286678 A081479

Adjacent sequences: A093425 A093426 A093427 * A093429 A093430 A093431

Keyword

nonn

Author

Lior Manor, May 12 2004

Extensions

More terms from Alexander Adamchuk, Aug 09 2006

Status

approved