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Carol numbers
Carol numbers are near-squares of the form  $(2^k-1)^2-2=4^k-2^{k+1}-1$.

The first Carol numbers are 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, 4190207, 16769023, 67092479, 268402687 more terms

The first Carol numbers which happen to be prime are 7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799.

Currently the largest Carol prime known is  $(2^{253987}-1)^2-2$, a number of 152916 digits discovered by C.Emmanuel.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of Carol numbers

A graph displaying how many Carol numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of Carol numbers
Useful links Mathworld, Carol Number
Wikipedia, Carol number
OEIS, Sequence A093112
OEIS, Sequence A091515
Carol numbers can also be... (you may click on names or numbers)

aban 47 223 959 alternating 47 16127 apocalyptic 3967 16127 arithmetic 47 223 959 3967 16127 65023 261119 1046527 4190207 canyon 959 65023 Chen 47 16127 congruent 47 223 959 3967 16127 65023 261119 1046527 4190207 cyclic 47 223 959 3967 16127 261119 1046527 de Polignac 959 3967 65023 deficient 47 223 959 3967 16127 65023 261119 1046527 4190207 dig.balanced 16127 Duffinian 959 65023 261119 4190207 economical 47 223 3967 16127 1046527 4190207 16769023 emirp 16127 1046527 16769023 equidigital 47 223 3967 16127 1046527 4190207 16769023 fibodiv 47 good prime 223 happy 16127 Honaker 1046527 iban 47 223 junction 65023 lonely 3967 Lucas 47 lucky 223 magnanimous 47 metadrome 47 oban 959 odious 47 223 959 3967 16127 65023 261119 1046527 4190207 16769023 67092479 268402687 palindromic 959 pernicious 47 223 3967 16127 261119 1046527 plaindrome 47 223 prime 47 223 3967 16127 1046527 16769023 1073676287 68718952447 274876858367 repfigit 47 self 261119 16769023 semiprime 959 261119 67092479 sphenic 4190207 strong prime 223 16127 super-d 261119 truncatable prime 47 223 3967 uban 47 Ulam 47 undulating 959 wasteful 959 65023 261119 weak prime 47 3967 1046527 16769023