A023811 Largest metadrome (number with digits in strict ascending order) in base n.

0, 1, 5, 27, 194, 1865, 22875, 342391, 6053444, 123456789, 2853116705, 73686780563, 2103299351334, 65751519677857, 2234152501943159, 81985529216486895, 3231407272993502984, 136146740744970718253, 6106233505124424657789, 290464265927977839335179

Offset

1,3

Comments

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

The smallest permutational number in A134640 in the n-positional system. - Artur Jasinski, Nov 07 2007

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.

Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Formula

a(n) = Sum_{j=1...n-1} j*n^(n-1-j).

lim_{n->infinity} a(n)/a(n-1) - a(n-1)/a(n-2) = exp(1). - Conjectured by Gerald McGarvey, Sep 26 2004. Follows from the formula below and lim_{n->infinity} (1+1/n)^n = e. - Franklin T. Adams-Watters, Jan 25 2010

a(n) = (n^n-n^2+n-1)/(n-1)^2 = A058128(n)-1 = n*A060073(n)-1 (for n>=2). - Henry Bottomley, Feb 21 2001

Example

a(5) = 1234[5] (in base 5) = 1*5^3 + 2*5^2 + 3*5 + 4 = 125 + 50 + 15 + 4 = 194.
a(10) = 123456789 (in base 10).

Maple

0, seq((n^n-n^2+n-1)/(n-1)^2, n=2..100); # Robert Israel, Dec 13 2015

Mathematica

Table[Total[(#1 n^#2) & @@@ Transpose@ {Range[n - 1], Reverse@ (Range[n - 1] - 1)}], {n, 20}] (* Michael De Vlieger, Jul 24 2015 *)
Table[Sum[(b - k)*b^(k - 1), {k, b - 1}], {b, 30}] (* Clark Kimberling, Aug 22 2015 *)
Table[FromDigits[Range[0, n - 1], n], {n, 20}] (* L. Edson Jeffery, Dec 13 2015 *)

Prog

(PARI) {for(i=1, 18, cuo=0; for(j=1, i-1, cuo=cuo+j*i^(i-j-1)); print1(cuo, ", "))} \\\ Douglas Latimer, May 16 2012
(PARI) A023811(n)=if(n>1, (n^n-n^2)\(n-1)^2+1)  \\ M. F. Hasler, Jan 22 2013
(Magma) [0] cat [(n^n-n^2+n-1)/(n-1)^2: n in [2..20]]; // Vincenzo Librandi, May 22 2012
(Haskell)
a023811 n = foldl (\val dig -> val * n + dig) 0 [0 .. n - 1]
-- Reinhard Zumkeller, Aug 29 2014
(Python)
def a(n): return (n**n - n**2 + n - 1)//((n - 1)**2) if n > 1 else 0
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Apr 24 2023

Crossrefs

Cf. A049363, A051846, A058128, A060073.

Cf. A062813, A134640.

Sequence in context: A326094 A232683 A240637 * A126119 A105631 A167019

Adjacent sequences: A023808 A023809 A023810 * A023812 A023813 A023814

Keyword

nonn,easy,base

Author

Olivier Gérard

Extensions

Edited by M. F. Hasler, Jan 22 2013

Status

approved