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

1, 43, 2462, 140081, 20338085, 2610787117

Offset: 1

Author

Joonas Pohjonen, Aug 28 2014

Keywords

• nonn
• base
• hard
• more

Comments

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

a(n) >= A049363(n+1) - 1 for all n. - Derek Orr, Aug 31 2014

From Derek Orr, Aug 31 2014 (Start):

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

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.

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.

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.

Generalizing... (Conjecture)

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.

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

(End)

a(1)-a(6) are confirmed for all n <= 10^11. - Hiroaki Yamanouchi, Sep 21 2014

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.

a(7) >= 314941024802. - Hiroaki Yamanouchi, Sep 21 2014

Examples

			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.
		

Crossrefs

Cf. A037968.

Extensions

a(6) from Hiroaki Yamanouchi, Sep 21 2014