A Kynea number is an integer of the form
An equivalent formula is
This indicates that a Kynea number is the nth 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:
- 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, ... (sequence A093069 in the 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:
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 cannot 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 (sequence A091514 in the OEIS).
As of 2006, the largest known prime Kynea number has index n = 281621 and approximately equals 5.5×10169552. It was found by Cletus Emmanuel in November 2005, using k-Sieve from Phil Carmody and OpenPFGW. This is the 46th Kynea prime.