A249626 - OEIS

%I #11 Apr 11 2016 08:46:38

%S 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,

%T 29,30,34,35,36,37,38,39,40,45,46,47,48,49,50,56,57,58,59,60,67,68,69,

%U 70,78,79,80,89,90,100,21,31,41,51,61,71,81,91,22,32,42

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

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

%C 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;

%C A249648 gives positions of terms containing a zero.

%H Reinhard Zumkeller, <a href="/A249626/b249626.txt">Table of n, a(n) for n = 0..10000</a>

%e 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.

%e 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;

%e 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.

%o (Haskell)

%o import Data.List (delete, group, sortBy); import Data.Function (on)

%o a249626 n = a249626_list !! n

%o a249626_list = f (zip [0,0..] [0..9]) a031298_tabf where

%o    f acds@((_,dig):_) zss = g zss where

%o      g (ys:yss) = if dig `elem` ys

%o                      then y : f acds' (delete ys zss) else g yss

%o        where y = foldr (\d v -> 10 * v + d) 0 ys

%o              acds' = sortBy (compare `on` fst) $

%o                     addd (sortBy (compare `on` snd) acds)

%o                          (sortBy (compare `on` snd) $

%o                                  zip (map length gss) (map head gss))

%o              addd cds [] = cds

%o              addd []   _ = []

%o              addd ((c, d) : cds) yys'@((cy, dy) : yys)

%o                   | d == dy  = (c + cy, d) : addd cds yys

%o                   | otherwise = (c, d) : addd cds yys'

%o              gss = sortBy compare $ group ys

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

%K nonn,look,base

%O 0,3

%A _Reinhard Zumkeller_, Nov 03 2014