Carol number

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

Carol numbers were first studied by Cletus Emmanuel, who named them after a friend, Carol G. Kirnon.

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

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

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