A037968 -id:A037968 - cp's OEIS frontend

A049363 a(1) = 1; for n > 1, smallest digitally balanced number in base n.

1, 2, 11, 75, 694, 8345, 123717, 2177399, 44317196, 1023456789, 26432593615, 754777787027, 23609224079778, 802772380556705, 29480883458974409, 1162849439785405935, 49030176097150555672, 2200618769387072998445, 104753196945250864004691, 5271200265927977839335179

Offset: 1

Harvey P. Dale

A037968(a(n)) = n and A037968(m) < n for m < a(n). - Reinhard Zumkeller, Oct 27 2003

Also smallest pandigital number in base n. - Franklin T. Adams-Watters, Nov 15 2006

			a(6) = 102345_6 = 1*6^5 + 2*6^3 + 3*6^2 + 4*6^1 + 5*6^0 = 8345.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Pandigital Number
Wikipedia, Pandigital number
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Column k=1 of A061845 and A378000 (for n>1).

Cf. A031443, A049354, A049355, A023811.

a049363 n = foldl (\v d -> n * v + d) 0 (1 : 0 : [2..n-1])
-- Reinhard Zumkeller, Apr 04 2012

a:= n-> n^(n-1)+add((n-i)*n^(i-1), i=1..n-2):
seq(a(n), n=1..23);  # Alois P. Heinz, May 02 2020

Table[FromDigits[Join[{1,0},Range[2,n-1]],n],{n,20}] (* Harvey P. Dale, Oct 12 2012 *)

A049363(n)=n^(n-1)+sum(i=1,n-2,n^(i-1)*(n-i))  \\ M. F. Hasler, Jan 10 2012

A049363(n)=if(n>1,(n^n-n)/(n-1)^2+n^(n-2)*(n-1)-1,1)  \\ M. F. Hasler, Jan 12 2012

def A049363(n): return (n**n-n)//(n-1)**2+n**(n-2)*(n-1)-1 if n>1 else 1 # Chai Wah Wu, Mar 13 2024

a(n) = (102345....n-1) in base n. - Ulrich Schimke (ulrschimke(AT)aol.com)
For n > 1, a(n) = (n^n-n)/(n-1)^2 + n^(n-2)*(n-1) - 1 = A023811(n) + A053506(n). - Franklin T. Adams-Watters, Nov 15 2006
a(n) = n^(n-1) + Sum_{m=2..n-1} m * n^(n - 1 - m). - Alexander R. Povolotsky, Sep 18 2022

More terms from Ulrich Schimke (ulrschimke(AT)aol.com)

A037914 a(n) is the least base b>=2 such that the number of distinct digits in the base b representation of n is maximized.

Original entry on oeis.org

2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 4, 3, 2, 3, 2, 2, 4, 2, 2, 4, 4, 3, 4, 3, 3, 3, 3, 3, 4, 2, 3, 4, 5, 6, 3, 2, 4, 3, 3, 3, 3, 4, 4, 3, 4, 5, 4, 3, 4, 3, 3, 3, 6, 3, 6, 3, 3, 3, 3, 4, 6, 3, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 5, 6, 4, 3, 5, 5, 3, 3, 3, 3, 6

Offset: 1

Clark Kimberling

Original title: a(n)=least base b>=2 such that f(b,n)>=f(b',n) for all b'>=2, where f(b,n)=number of distinct base b digits of n.

Sean A. Irvine, Java program (github).

See A037968 for the greatest number of digits over all bases b>=2.

Offset corrected and title simplified by Sean A. Irvine, Dec 27 2020

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

Original entry on oeis.org

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

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A049363 a(1) = 1; for n > 1, smallest digitally balanced number in base n.

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A037914 a(n) is the least base b>=2 such that the number of distinct digits in the base b representation of n is maximized.

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

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