A Kynea number
is an integer of the form
- <math>4^n + 2^ - 1</math>.
An equivalent formula is
- <math>(2^n + 1)^2 - 2</math>.
This indicates that a Kynea number is the n
th power of 4 plus the (n
+ 1)th Mersenne number
.Kynea numbers were studied by Cletus Emmanuel who named them after a baby girl.
The sequence of Kynea numbers starts with:
- 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, ... .
The binary representation
of the n
th Kynea number is a single leading one, followed by n
- 1 consecutive zeroes, followed by n
+ 1 consecutive ones, or to put it algebraically:
- <math>4^n + sum_^n 2^i.</math>
So, for example, 23 is 10111 in binary, 79 is 1001111, etc. The difference between the n
th Kynea number and the n
th Carol number
is the (n
+ 2)th power of two
Prime Kynea numbers
Starting with 7, every third Kynea number is a multiple of 7. Thus, for a Kynea number to be a prime number
, its index n
can not be of the form 3x
+ 1 for x
> 0. The first few Kynea numbers that are also prime are 7, 23, 79, 1087, 66047, 263167, 16785407 .
As of 2006, the largest known prime Kynea number has index n
= 281621 and approximately equals 5.5×10<sup>169552</sup>. It was found by Cletus Emmanuel in November 2005, using k
-Sieve from Phil Carmody and OpenPFGW. This is the 46th Kynea prime.