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.^[1]

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).

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.

References

1. Cletus Emmanuel's statement on Yahoo group PrimeNumbers

External links

• Weisstein, Eric W. "Near-Square Prime". MathWorld.

• Prime Database entry for Kynea(281621)