Kynea number

A Kynea number is an integer of the form

:$4^n\; +\; 2^\{n\; +\; 1\}\; -\; 1$.

An equivalent formula is

:$(2^n\; +\; 1)^2\; -\; 2$.

This indicates that a Kynea number is the "n"th power of 4 plus the ("n" + 1)th Mersenne number. The first few Kynea numbers are

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407 OEIS|id=A093069

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:

:$4^n\; +\; sum\_\{i\; =\; 0\}^n\; 2^i.$

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.

Starting with 7, every third Kynea number is a multiple of 7. Thus, for a Kynea number to also 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 (these are listed in Sloane's OEIS2C|id=A091514). As of 2006, the largest known Kynea number that is also a prime is the Kynea number for "n" = 281621, approximately 5.455289117190661 × 10¹⁶⁹⁵⁵². It was found by Cletus Emmanuel in November 2005, using k-Sieve from Phil Carmody and OpenPFGW. This is the 46th Kynea prime. Kynea numbers were studied by Cletus Emmanuel who arbitrarily named them after a baby girl. [http://tech.groups.yahoo.com/group/primenumbers/message/14584]

External links

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* [http://primes.utm.edu/primes/page.php?id=75878 Prime Database entry for Kynea(281621)]