A316272 A fractal-like sequence: erasing all pairs of consecutive terms that include a prime and a composite number (in any order) leaves the sequence unchanged.

1, 2, 3, 4, 1, 6, 5, 2, 3, 7, 8, 4, 1, 6, 9, 11, 5, 2, 3, 7, 13, 10, 8, 4, 1, 6, 9, 12, 17, 11, 5, 2, 3, 7, 13, 19, 14, 10, 8, 4, 1, 6, 9, 12, 15, 23, 17, 11, 5, 2, 3, 7, 13, 19, 29, 16, 14, 10, 8, 4, 1, 6, 9, 12, 15, 18, 31, 23, 17, 11, 5, 2, 3, 7, 13, 19, 29, 37, 20, 16, 14, 10, 8, 4, 1

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jun 28 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 either with the smallest prime P > 2 not yet present inside another pair of parentheses or with the smallest composite C > 1 not yet present inside another pair of parentheses ;

3) always end the content inside a pair of parentheses either with the smallest composite C > 1 not yet present inside another pair of parentheses or with the smallest prime > 2 not yet present inside another pair of parentheses;

4) after a(1) = 1 and a(2) = 2, always try to extend the sequence with a duplicate > 1 of the oldest term of the sequence not yet duplicated; if this leads to a contradiction, open a new pair of parentheses.

Examples

			Parentheses are added around each pair of terms made of a composite and a prime number (in any order):
(1,2),(3,4),1,(6,5),2,3,(7,8),4,1,6,(9,11),5,2,3,7,(13,10),8,4,1,6,9,(12,17),11,...
Erasing all the parenthesized contents yields
(...),(...),1,(...),2,3,(...),4,1,6,(....),5,2,3,7,(.....),8,4,1,6,9,(.....),11,...
We see that the remaining terms rebuild the starting sequence.
		

Links

• Eric Angelini, Table of n, a(n) for n = 1..20706

Crossrefs

For other "erasing criteria", see A303845 (prime by concatenation), A274329 (pair summing up to a prime), A303936 (pair not summing up to a prime), A303948 (pair sharing a digit), A302389 (pair having no digit in common), A303950 (pair summing up to a Fibonacci), A303951 (pair not summing up to a Fibonacci), A303953 (pair summing up to a square), A303954 (pair not summing up to a square).