Kynea number

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Description:
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 nth 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, ... .


Properties

The binary representation of the nth 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 nth Kynea number and the nth 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.





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