A303948 A fractal-like sequence: erasing all pairs of consecutive terms that have at least one digit in common leaves the sequence unchanged.

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

Offset: 1

Author

Eric Angelini and Lars Blomberg, May 03 2018

Keywords

• nonn
• base

Comments

The sequence is fractal-like as it embeds an infinite number of copies of itself.

The sequence was built according to these rules (see, in the Example section, the parenthesization technique):

1) no overlapping pairs of parentheses;

2) always start the content inside a pair of parentheses with the smallest integer S > 9 not yet present inside another pair of parentheses;

3) always end the content inside a pair of parentheses with the smallest integer H > 9 not yet present inside another pair of parentheses such that the integers S and H have at least one digit in common;

4) after a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5, a(6) = 6, a(7) = 7, a(8) = 8, a(9) = 9, a(10) = 10, always try to extend the sequence with a duplicate > 9 of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses;

5) Never use a term of A171102 (Pandigital numbers: numbers containing the digits 0-9. Version 2: each digit appears at least once).

Examples

			Parentheses are added around each pair of terms having at least one digit in common:
1,2,3,4,5,6,7,8,9,(10,11),(20,12),(30,13),(22,21),(33,23),10,(24,14),(25,15),(26,16),(27,17),(28,18),(29,19),(32,31),(40,34),11,20,(35,36),12,30,(41,42),13,
Erasing all the parenthesized contents yields
1,2,3,4,5,6,7,8,9,(.....),(.....),(.....),(.....),(.....),10,(.....),(.....),(.....),(.....),(.....),(.....),(.....),(.....),11,20,(.....),12,30,(.....),13,
We see that the remaining terms slowly rebuild the starting sequence.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A303845 for another "erasing criterion" (prime by concatenation).