Kynea number

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A Kynéa number (pronounced: Ka-Nay) 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 Kynéa number is the nth power of 4 plus the (n + 1)th Mersenne number. The first few Kynéa numbers are

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407 (sequence A093069 in OEIS)

The binary representation of the nth Kynéa 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 nth Kynéa number and the nth Carol number is the (n + 2)th power of two.

Starting with 7, every third Kynéa number is a multiple of 7. Thus, for a Kynéa number to also be a prime number, its index n can not be of the form 3x + 1 for x > 0. The first few Kynéa numbers that are also prime are 7, 23, 79, 1087, 66047, 263167, 16785407 (these are listed in Sloane's A091514). As of 2006, the largest known Kynéa number that is also a prime is the Kynéa number for n = 281621, approximately 5.455289117190661 × 10¹⁶⁹⁵⁵². It was found by Cletus Emmanuel in November of 2005, using k-Sieve from Phil Carmody and OpenPFGW. This is the 46th Kynéa prime. These numbers were first encountered in 1994 by Cletus Emmanuel.