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Numbers k > 1 such that the ratio (numbers of zeros)/(total length) in the binary representation of k^2 is a new minimum.
2

%I #14 Nov 30 2022 14:55:23

%S 2,3,5,11,45,181,48589783221,66537313397,398064946368587,

%T 796095014224053

%N Numbers k > 1 such that the ratio (numbers of zeros)/(total length) in the binary representation of k^2 is a new minimum.

%e k k^2 (binary zeros)/A070939(k^2)

%e . . . k^2 written in binary

%e 2 4 2/3 [1, 0, 0]

%e 3 9 1/2 [1, 0, 0, 1]

%e 5 25 2/5 [1, 1, 0, 0, 1]

%e 11 121 2/7 [1, 1, 1, 1, 0, 0, 1]

%e 45 2025 3/11 [1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1]

%e 181 32761 2/15 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1]

%Y Cf. A000120, A000290, A070939, A159918, A230097.

%K nonn,base,hard,more

%O 1,1

%A _Hugo Pfoertner_, Oct 01 2022

%E a(7)-a(8) from _Michael S. Branicky_, Oct 01 2022 using A230097, verified with exhaustive search Oct 02 2022

%E a(9)-a(10) from _Hugo Pfoertner_, Nov 30 2022