Carol number

English

Etymology

First studied by Cletus Emmanuel, who named it after a friend, Carol G. Kirnon.

Noun

Carol number (plural Carol numbers)

1. (mathematics) An integer of the form $4^{n}-2^{n+1}-1$.
   • 2009, The Philosophers' Magazine - Issue 47, page 4:

     Mathematically, it is actually interesting in all sorts of ways, being both a Keith number and a Carol number.

   • 2014, Rajesh Kumar Thakur, The Power of Mathematical Numbers, page 131:

     […] 7. There are 47 occurrences of 47 in the first thousand prime numbers. 8. The atomic number of Silver is 47. 9. It is a Carol number. […]

   • 2019, Dr. Ir. Maarten Looijen, Over getallen gesproken - Talking about numbers, page 70:

     In 2007 the largest Carol number was found for n = 253987, consisting of 152916 digits, by using the programs MultiSieve and PrimeFormGW.

   • 2021, Elena Deza, Mersenne Numbers And Fermat Numbers, page 136:

     For $n>2,$ the binary representation of the $n$-th Carol number is $n-2$ consecutive unities, a single zero in the middle, and $n+1$ more consecutive unities, or to put it algebraically, $\textstyle {CN(n)=\sum _{i=1,i\neq n+2}^{2n}2^{i-1}}.$