A306424 Numbers k such that the base-b expansion of k for each b = 3..k-1 never contains more than two distinct digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 20, 22, 23, 25, 26, 31, 37, 43

Offset

1,2

Comments

Conjecture: The sequence is finite, with 43 being the last term.

I checked the conjecture to 10809638.

This was proved by an autonomous AI agent, see the Lean file. The proof uses two regimes: for 43 < k <= 288, computation brute-forces every case; for k >= 289, it locates the base b = floor(sqrt(k)), writes k as c^2 + y*c + z where y is a fixed per-case constant and z is either a fixed constant or a linear function of the remainder r = k - b^2. - Ralf Stephan, Jun 01 2006

Links

Table of n, a(n) for n=1..23.

Google Deepmind, AlphaProof Nexus: A306424 Lean file

Example

10 is a term of the sequence, since the base-b expansions of 10 for b = 3..9 are 101, 22, 20, 14, 13, 12, 11, respectively, and none of those expansions contain more than two distinct digits.

Mathematica

Select[Range@ 100, Max@ Table[Length@ Union@ IntegerDigits[#, b], {b, 3, # - 1}] <= 2 &] (* Michael De Vlieger, Feb 15 2019 *)

Prog

(PARI) is(n) = for(b=3, n-1, my(d=digits(n, b)); if(#vecsort(d, , 8) > 2, return(0))); 1

Crossrefs

Sequence in context: A154314 A239348 A191881 * A005524 A191890 A247814

Adjacent sequences: A306421 A306422 A306423 * A306425 A306426 A306427

Keyword

nonn,base,more

Author

Felix Fröhlich, Feb 14 2019

Status

approved