Kynea number

A Kynea number is an integer of the form

[math]\displaystyle{ 4^n + 2^{n + 1} - 1 }[/math].

An equivalent formula is

[math]\displaystyle{ (2^n + 1)^2 - 2 }[/math].

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:

[math]\displaystyle{ 4^n + \sum_{i = 0}^n 2^i. }[/math]

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.

External links

• Weisstein, Eric W.. "Near-Square Prime". http://mathworld.wolfram.com/Near-SquarePrime.html. 
• Prime Database entry for Kynea(661478)
• Carol and Kynea Primes
• Carol and Kynea Prime Search
• Carol-Kynea prime in Prime wiki