Carol number

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A Carol number is an integer of the form $4 n - 2 n + 1 - 1$ . An equivalent formula is $(2 n - 1) 2 - 2$ . The first few Carol numbers are: −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527 (sequence A093112 in OEIS).

For n > 2, the binary representation of the nth Carol number is n − 2 consecutive ones, a single zero in the middle, and n + 1 more consecutive ones, or to put it algebraically,

$\sum_{i \ne n + 2}^{2n} 2^{i - 1}.$

So, for example, 47 is 101111 in binary, 223 is 11011111, etc. The difference between the nth Mersenne number and the nth Carol number is $2 n + 1$ . This gives yet another equivalent expression for Carol numbers, $(2 2 n - 1) - 2 n + 1$ . The difference between the nth Kynea number and the nth Carol number is the (n + 2)th power of two.

Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a prime number, its index n cannot be of the form 3x + 2 for x > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127 (these are listed in Sloane's A091516). As of July 2007, the largest known Carol number that is also a prime is the Carol number for n = 253987.[1] It was found by Cletus Emmanuel in May 2007, using the programs MultiSieve and PrimeFormGW. It is the 40th Carol prime.

Carol numbers and the formula to discover them were first found and studied by Cletus Emmanuel who named them after a friend, Carol G. Kirnon[2][3]