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).
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.[2][3] It was found by Ryan Propper in July 2019 using the programs CKSieve and PrimeFormGW. It is the 51st Kynea prime.
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References
"Yahoo! Groups". groups.yahoo.com. Retrieved 2020-08-10.
Entry for 852770th Kynea number at Prime Pages
Carol and Kynea Prime Search by Mark Rodenkirch
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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