A188381 Negabinary Keith numbers.

2, 3, 4, 7, 9, 13, 16, 36, 55, 64, 162, 256, 458, 1024, 1829, 4096, 7316, 15119, 16384, 18970, 37702, 37723, 45171, 60476, 65536, 84506, 262144, 277263, 1048576, 1109052, 1722002, 2160570, 4194304, 10549178, 12699958, 15084573, 16777216, 31921069, 67108864

Offset: 1

Author

Alonso del Arte, Mar 29 2011

Keywords

• nonn
• base

Comments

Keith numbers are described in A007629. All powers of 4 appear. However, 2 is the only number of the form 2^n with n odd that appears in the sequence. That's because in negabinary, such numbers are represented as 11 followed by n 0's, and that leads to the sequence 1, 1, 0, ... , 0, 2, 3, 5, 10, 20, 40, 80, 160, ... up to 5(2^(n - 2)), and 5(2^(n - 2)) > 2^(n - 1). (See A020714).

Crossrefs

Cf. A007629, A039724, A162724.

Programs

Mathematica

(* First run the program from A039724 to define ToNegaBases *) keithFromListQ[n_Integer, digits_List] := Module[{seq = digits, curr = digits[[-1]], ord = Length[digits]}, While[curr < n, curr = Plus@@Take[seq, -ord]; AppendTo[seq, curr]]; Return[seq[[-1]] == n]]; Select[Range[2, 32768], keithFromListQ[#, IntegerDigits[ToNegaBases[#, 2]]] &]

Extensions

a(33)-a(39) from Amiram Eldar, Jan 29 2020