A249626 a(0) = 0, a(n+1) = smallest number, not occurring earlier, containing the smallest of the least frequently occurring digits in all preceding terms.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 34, 35, 36, 37, 38, 39, 40, 45, 46, 47, 48, 49, 50, 56, 57, 58, 59, 60, 67, 68, 69, 70, 78, 79, 80, 89, 90, 100, 21, 31, 41, 51, 61, 71, 81, 91, 22, 32, 42

Offset: 0

Author

Reinhard Zumkeller, Nov 03 2014

Keywords

• nonn
• look
• base

Comments

a(n) = A102823(n) for n <= 55;

not all numbers occur: all repunits (A002275) greater than 1 are missing; idea of proof: for n > 1 the digit 1 will never again be the smallest of least frequently occurring digits;

A249648 gives positions of terms containing a zero.

Examples

			n = 11: digits 0 and 1 occur twice in {a(k): k=0..10}, all other digits exactly once, where 2 is the smallest; therefore a(11) must contain digit 2, and 12 is the smallest unused number containing 2, hence a(11) = 12.
n = 55: digits 0..9 occur exactly 10 times in {a(k): k=0..54}; therefore a(55) must contain digit 0, the smallest digit; a(55) = 100, as 100 is the smallest unused number containing 0;
n = 56: least occurring digits in {a(k): k=0..10} are 2..9 and 2 is the smallest; therefore a(56) must contain digit 2, and 21 is the smallest unused number containing 2, hence a(56) = 21.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A031298, A002275, A011540, A249648, A102823.

Programs

Haskell

import Data.List (delete, group, sortBy); import Data.Function (on)
a249626 n = a249626_list !! n
a249626_list = f (zip [0,0..] [0..9]) a031298_tabf where
   f acds@((,dig):) zss = g zss where
     g (ys:yss) = if dig `elem` ys
                     then y : f acds' (delete ys zss) else g yss
       where y = foldr (\d v -> 10 * v + d) 0 ys
             acds' = sortBy (compare `on` fst) $
                    addd (sortBy (compare `on` snd) acds)
                         (sortBy (compare `on` snd) $
                                 zip (map length gss) (map head gss))
             addd cds [] = cds
             addd []   _ = []
             addd ((c, d) : cds) yys'@((cy, dy) : yys)
                  | d == dy  = (c + cy, d) : addd cds yys
                  | otherwise = (c, d) : addd cds yys'
             gss = sortBy compare $ group ys