Kynea Number

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A **Kynea number** is an integer of the form

An equivalent formula is

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:

## Properties

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:

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 3*x* + 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.

## References

## External links

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- <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

The sequence of Kynea numbers starts with:

- <math>4^n + sum_^n 2^i.</math>

So, for example, 23 is 10111 in binary, 79 is 1001111, etc. The difference between the

As of 2006, the largest known prime Kynea number has index

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