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Kynea numbers
A number of the form 4k + 2k+1 - 1. more

The Kynea numbers up to 1015 :

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 30000 values, from 2 to 1.567⋅1018061).

n\r 0  1 
2129999 2 
301500015000 3 
400129999 4 
5001500075007500 5 
601500010014999 6 
7100000200000000 7 
8001000029999 8 
90050000050000150005000 9 
1000175000001499907500 10 
1106000600060000006000030003000

A pictorial representation of the table above
motab
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.