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 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).

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:

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

Prime Kynea numbers

 Kynea numbers n Decimal Binary 1 7 111 2 23 10111 3 79 1001111 4 287 100011111 5 1087 10000111111 6 4223 1000001111111 7 16639 100000011111111 8 66047 10000000111111111 9 263167 1000000001111111111

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).

Their n values are: 1, 2, 3, 5, 8, 9, 12, 15, 17, 18, 21, 23, 27, 32, 51, 65, 87, 180, 242, 467, ... (sequence A091513 in the OEIS).

As of July 2019, 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.