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A246535 Largest number with at most n distinct digits in any base b >= 2 (written in decimal). 1

%I

%S 1,43,2462,140081,20338085,2610787117

%N Largest number with at most n distinct digits in any base b >= 2 (written in decimal).

%C a(n) is the last occurrence of n in A037968.

%C a(n) >= A049363(n+1) - 1 for all n. - _Derek Orr_, Aug 31 2014

%C From _Derek Orr_, Aug 31 2014 (Start):

%C At least for 1 <= n <= 5, a(n)+1 fails when written in base n^2+1. Examples:

%C a(1) = 1 written in base 2 is 1 (1 distinct digit). 2 written in base (2-1)^2+1 = 2 is 10. Thus 2 fails.

%C a(2) = 43 written in base 3 is 1121 (2 distinct digits). 44 written in base 2^2+1 = 5 is 134. Thus 44 fails.

%C a(3) = 2462 written in base 4 is 212132 (3 distinct digits). 2463 written in base 3^2+1 = 10 is 2463. Thus 2463 fails.

%C Generalizing... (Conjecture)

%C a(n) written in base n+1 has n distinct digits. a(n)+1 written in base n^2+1 will always have n+1 distinct digits.

%C Further, for 1 < n <= 5, a(n)-1 fails when written in base n^2+1.

%C (End)

%C a(1)-a(6) are confirmed for all n <= 10^11. - _Hiroaki Yamanouchi_, Sep 21 2014

%C a(6) = 2610787117 written in base 7 is 121461216151 (5 distinct digits), and 2610787118 written in base 6^2+1 = 37 is (1)(0)(24)(1)(22)(2)(0) (5 distinct digits). Therefore, Derek Orr's conjecture seems to be wrong.

%C a(7) >= 314941024802. - _Hiroaki Yamanouchi_, Sep 21 2014

%e a(2) = 43 since 43 has two distinct digits in bases 2 <= b <= 5, 7 <= b <= 41 and b = 43, and one distinct digit in bases b = 6, b = 42 and b >= 44. All greater numbers have at least 3 distinct digits in some base b >= 2.

%Y Cf. A037968.

%K nonn,base,hard,more

%O 1,2

%A _Joonas Pohjonen_, Aug 28 2014

%E a(6) from _Hiroaki Yamanouchi_, Sep 21 2014

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Last modified June 20 14:11 EDT 2021. Contains 345165 sequences. (Running on oeis4.)