A262323 Lexicographically earliest sequence of distinct terms such that the decimal representations of two consecutive terms overlap.

1, 10, 11, 12, 2, 20, 22, 21, 13, 3, 23, 30, 33, 31, 14, 4, 24, 32, 25, 5, 15, 41, 16, 6, 26, 42, 27, 7, 17, 51, 18, 8, 28, 52, 29, 9, 19, 61, 36, 43, 34, 40, 44, 45, 50, 35, 53, 37, 63, 38, 73, 39, 83, 48, 54, 46, 60, 56, 55, 57, 65, 58, 75, 47, 64, 49, 74

Offset: 1

Author

Paul Tek, Sep 19 2015

Keywords

• nonn
• look
• base
• nice

Comments

Two terms are said to overlap:

- if the decimal representation of one term is contained in the decimal representation of the other term (for example, 12 and 2 overlap),

- or if, for some k>0, the first k decimal digits (without leading zero) of one term correspond to the k last decimal digits of the other term (for example, 1017 and 1101 overlap).

This sequence is a permutation of the positive integers, with inverse A262255.

The first overlap involving 1 digit occurs between a(1)=1 and a(2)=10.

The first overlap involving 2 digits occurs between a(108)=100 and a(109)=110.

The first overlap involving 3 digits occurs between a(1039)=1017 and a(1040)=1101.

The first overlap involving 4 digits occurs between a(10584)=10212 and a(10585)=11021.

Examples

			The first terms of the sequence are:
+----+---------+
| n  | a(n)    |
+----+---------+
|  1 |  1      |
|  2 |  10     |
|  3 | 11      |
|  4 |  12     |
|  5 |   2     |
|  6 |   20    |
|  7 |  22     |
|  8 |   21    |
|  9 |    13   |
| 10 |     3   |
| 11 |    23   |
| 12 |     30  |
| 13 |    33   |
| 14 |     31  |
| 15 |      14 |
| 16 |       4 |
| 17 |      24 |
| 18 |     32  |
| 19 |      25 |
| 20 |       5 |
+----+---------+
		

Links

• Paul Tek, Table of n, a(n) for n = 1..10000
• Paul Tek, PERL program for this sequence
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A076654, A262255, A262283.

Cf. A262367 (fixed points), A262411 (ternary version), A262460 (hexadecimal version).

Programs

Haskell

import Data.List (inits, tails, intersect, delete)
a262323 n = a262323_list !! (n-1)
a262323_list = 1 : f "1" (map show [2..]) where
   f xs zss = g zss where
     g (ys:yss) | null (intersect its $ tail $ inits ys) &&
                  null (intersect tis $ init $ tails ys) = g yss
                | otherwise = (read ys :: Int) : f ys (delete ys zss)
     its = init $ tails xs; tis = tail $ inits xs
-- Reinhard Zumkeller, Sep 21 2015

Perl

See Links section.

Python

def overlaps(a, b):
  s, t = sorted([str(a), str(b)], key = lambda x: len(x))
  if any(t.startswith(s[i:]) for i in range(len(s))): return True
  return any(t.endswith(s[:i]) for i in range(1, len(s)+1))
def aupto(nn):
  alst, aset = [1], {1}
  for n in range(2, nn+1):
    an = 1
    while True:
      while an in aset: an += 1
      if overlaps(an, alst[-1]): alst.append(an); aset.add(an); break
      an += 1
  return alst
print(aupto(67)) # Michael S. Branicky, Jan 10 2021