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 (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[edit]
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 (n + 2)th power of two.
Prime Kynea numbers[edit]
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[update], 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[edit]
- ↑ "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
External links[edit]
- Weisstein, Eric W. "Near-Square Prime". MathWorld.
- Prime Database entry for Kynea(661478)
- Carol and Kynea Primes
- Carol and Kynea Prime Search
- Carol-Kynea prime in Prime wiki
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