A060857 Describe all the numbers already used (sorted into increasing order - not splitting numbers up into their digits).

1, 11, 31, 4113, 612314, 8112332416, 1113253342618, 131528344153628111, 1617210364354648211113, 181921239445661758110311213116, 2211121431146586276829210411112313216118

Offset: 0

Author

Henry Bottomley, May 03 2001

Keywords

• base
• nice
• nonn

Examples

			One; one one; three ones; four ones, one three; six ones, two threes, one four; eight ones, one two, three threes, two fours, one six; eleven ones, three twos, five threes, three fours, two sixes, one eight; thirteen [note not 15] ones, five twos, eight threes, four fours, one five, three sixes, two eights, one eleven [note than numbers >9 are preserved as wholes rather than as a collection of digits]; etc.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..60
• Onno M. Cain and Sela T. Enin, Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60, arXiv:2004.00209 [math.NT], 2020.

Crossrefs

This is a combination of methods used in A005151 and A045982. The first word of each term (the number of ones used earlier) seems to be equal to A030711 and A030761.

Programs

Haskell

import Data.List (group, sort, transpose)
a060857 n = a060857_list !! n
a060857_list = 1 : f [1] :: [Integer] where
   f xs = (read $ concatMap show ys) : f (xs ++ ys) where
          ys = concat $ transpose [map length zss, map head zss]
          zss = group $ sort xs
-- Reinhard Zumkeller, Jan 25 2014

Mathematica

FromDigits /@ Nest[Append[#, Flatten@ Map[Reverse, Tally@ Sort@ Flatten@ # ] ] &, {{1}}, 10] (* Michael De Vlieger, Jul 15 2020 *)

Python

def summarize_lst(lst):
  ans = []
  for d in sorted(set(lst)): ans += [lst.count(d), d]
  return ans
def aupton(nn):
  alst, arunninglst = [1], [1]
  for n in range(nn):
    nxt_lst = summarize_lst(arunninglst)
    arunninglst += nxt_lst
    alst.append(int("".join(map(str, nxt_lst))))
  return alst
print(aupton(10)) # Michael S. Branicky, Jan 11 2021