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

The first Kynea numbers are 2, 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247 more terms

The first Kynea numbers which happen to be prime are 2, 7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023

Currently the largest Kynea prime known is  $(2^{281621}-1)^2-2$, a number of 169553 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 Kynea numbers

A graph displaying how many Kynea numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of Kynea numbers
Useful links Mathworld, Kynea Number
Wikipedia, Kynea number
OEIS, Sequence A093069
OEIS, Sequence A091514
Kynea numbers can also be... (you may click on names or numbers)

aban 23 79 287 alternating 23 apocalyptic 287 1087 4223 16639 arithmetic 23 79 287 1087 4223 16639 66047 263167 1050623 4198399 brilliant 4198399 268468223 Chen 23 66047 congruent 23 79 287 1087 4223 16639 66047 263167 1050623 4198399 cyclic 23 79 287 1087 4223 16639 66047 263167 1050623 4198399 de Polignac 1087 263167 deficient 23 79 287 1087 4223 16639 66047 263167 1050623 4198399 Duffinian 4223 16639 1050623 4198399 economical 23 79 287 1087 16639 66047 263167 1050623 16785407 emirp 79 emirpimes 16639 equidigital 23 79 287 1087 16639 66047 263167 1050623 16785407 esthetic 23 evil 23 287 4223 66047 1050623 16785407 268468223 good prime 1087 263167 happy 23 79 iban 23 4223 lonely 23 lucky 79 1087 16639 m-pointer 23 magnanimous 23 metadrome 23 79 modest 23 79 mountain 287 oban 23 79 odious 79 1087 16639 263167 4198399 67125247 pancake 79 panconsummate 23 pentagonal 287 pernicious 79 1087 263167 4198399 plaindrome 23 79 prime 23 79 1087 66047 263167 16785407 1073807359 17180131327 68720001023 self 1087 semiprime 287 4223 16639 1050623 4198399 Sophie Germain 23 strong prime 79 1087 263167 16785407 super-d 1050623 truncatable prime 23 79 uban 23 79 wasteful 4223 4198399 weak prime 23 66047 Woodall 23