Kynea number


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 th Mersenne number.
Kynea numbers were studied by Cletus Emmanuel who named them after a baby girl.
The sequence of Kynea numbers starts with:

Properties

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:
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 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 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.
Their n values are: 1, 2, 3, 5, 8, 9, 12, 15, 17, 18, 21, 23, 27, 32, 51, 65, 87, 180, 242, 467,....
, the largest known prime Kynea number has index n = 852770, which has 513419 digits. It was found by Ryan Propper in July 2019 using the programs CKSieve and PrimeFormGW. It is the 51st Kynea prime.