Kynea number

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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 nth 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 (sequence A093069 in OEIS)

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:

$4^n + \sum_{i = 0}^n 2^i.$

So, for example, 23 is 10111 in binary, 79 is 1001111, etc. The difference between the nth Kynea number and the nth 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 3x + 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 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 of 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.[1]