A049363 -id:A049363 - cp's OEIS frontend

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

Joonas Pohjonen, Aug 28 2014

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

			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.

Cf. A037968.

a(6) from Hiroaki Yamanouchi, Sep 21 2014

A279087 Smallest number k such that k^j is (at least conjecturally) pandigital in base n for every j > 0.

Original entry on oeis.org

2, 15, 108, 694, 8415, 123759, 2178351, 44319300, 1023458769, 26432625775, 754777811227, 23609224082118, 802772380675044, 29480883459072073, 1162849439785537515, 49030176097152072920, 2200618769387075086589, 104753196945250866857691, 5271200265927977842382779

Offset: 2

Jon E. Schoenfield, Jan 28 2017

A049363(n), the smallest number that is pandigital in base n, provides a lower bound. A049363(n)^2 is also pandigital in base n at n = {2, 5, 6, 10, 11, 17}, but nowhere else up to 5000, and of these, A049363(n)^3 is pandigital in base n only at n = 2 and 5. A049363(2)^j = 2^j is clearly pandigital in base 2 for every j > 0 (as its binary expansion is simply a one followed by j zeros), and A049363(5)^j = 694^j seems nearly certain to be pandigital in base 5 for all j > 0. (That 694^j is pandigital in base 5 has been confirmed for every positive j up through 10^6.)

			a(3) = 15 because 15 = 120_3, 15^2 = 22100_3, 15^3 = 11122000_3, 15^4 = 2120110000_3, and (apparently) 15^j for all j > 0 are all pandigital in base 3 (15^j is pandigital in base 3 for every positive j up through at least 2*10^6), and no number smaller than 15 has this property. (E.g., A049363(3) = 11 = 102_3 is pandigital in base 3, but 11^2 = 11111_3 is not.)
a(5) = 694 because not only is 694 = 10234_5 pandigital in base 5 (it happens to be the smallest such number A049363(5)), but so are 694^2 = 110403021_5, 694^3 = 1141032133014_5, 694^4 = 12300040122031441_5, and (apparently) 694^j for every j > 0, and no number smaller than 694 has this property.

Cf. A049363.

cp's OEIS Frontend

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

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