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

:<math>4^n + 2^{n + 1} - 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.<ref>[http://tech.groups.yahoo.com/group/primenumbers/message/14584]</ref>

The sequence of Kynea numbers starts with:
: [[7 (number)|7]], [[23 (number)|23]], [[79 (number)|79]], 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, ... {{OEIS|id=A093069}}.

== Properties ==

The [[Binary numeral system|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_{i = 0}^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 3''x'' + 1 for ''x'' > 0. The first few Kynea numbers that are also prime are 7, 23, 79, 1087, 66047, 263167, 16785407 {{OEIS|id=A091514}}. 

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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==References==
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==External links==
* {{MathWorld|title=Near-Square Prime|urlname=Near-SquarePrime}}
* [http://primes.utm.edu/primes/page.php?id=75878 Prime Database entry for Kynea(281621)]

{{Prime number classes|state=collapsed}}
{{Classes of natural numbers}}

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[[Category:Integer sequences]]