A262557 Numbers with digits in strictly decreasing order, sorted lexicographically.

0, 1, 10, 2, 20, 21, 210, 3, 30, 31, 310, 32, 320, 321, 3210, 4, 40, 41, 410, 42, 420, 421, 4210, 43, 430, 431, 4310, 432, 4320, 4321, 43210, 5, 50, 51, 510, 52, 520, 521, 5210, 53, 530, 531, 5310, 532, 5320, 5321, 53210, 54, 540, 541, 5410, 542, 5420, 5421

Offset: 1

Author

N. J. A. Sloane, Oct 14 2015

Keywords

• nonn
• base
• fini
• full

Comments

Original name: "Countdown sequences, allowing gaps."

Only digits 0 through 9 are used. The last term is 9876543210.

Equals A009995, sorted lexicographically. - Reinhard Zumkeller, Oct 14 2015

There are 2^k terms starting with digit k >= 0, they start at index 2^k. The countdown sequences, i.e., digits of the n-th term, are given in rows of A272011. - M. F. Hasler, Dec 11 2019

References

• Donald S. McDonald, Email message to N. J. A. Sloane, Oct 14 2015.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1023

Crossrefs

Cf. A009995, A263327, A263328.

Programs

Haskell

a262557 n = a262557_list !! (n-1)
a262557_list = 0 : f [[0]] where
   f xss = if x < 9 then (map (read . concatMap show) zss) ++ f zss else []
           where zss = (map (z :) $ map tail xss) ++ (map (z :) xss)
                 z = x + 1; x = head $ head xss
-- Reinhard Zumkeller, Oct 14 2015

Mathematica

A262557[n_] := FromDigits[BitLength[n] - Flatten[Position[IntegerDigits[n, 2], 1]]]; Array[A262557, 100] (* or *)
A262557full = Rest[Map[FromDigits, LexicographicSort[Subsets[Range[9, 0, -1]]]]] (*  Paolo Xausa, Feb 13 2024 *)

PARI

is_A262557 = is_A009995
apply( A262557(n)=fromdigits(Vecrev(vecextract([0..exponent(n+!n)],n))), [1..99])
# A262557=concat(apply(x(i)=concat(vector(i%10+1,j,if(j>1,x(i*10+j-2),i))),[0..9])) \\ M. F. Hasler, Dec 11 2019

Python

from itertools import combinations
afull = list(map(int, sorted("".join(c) for i in range(1, 11) for c in combinations("9876543210", i)))) # Michael S. Branicky, Feb 13 2024

Formula

a(n) = A009995(A263328(n)); a(A263327(n)) = A009995(n). - Reinhard Zumkeller, Oct 15 2015

Extensions

New name from M. F. Hasler, Dec 11 2019