A303845 A fractal-like sequence: erasing all pairs of consecutive terms that produce a prime by concatenation leaves the sequence unchanged.

1, 2, 3, 2, 4, 7, 5, 9, 3, 2, 4, 6, 13, 8, 11, 7, 5, 10, 19, 12, 17, 14, 23, 15, 31, 9, 3, 2, 4, 6, 16, 21, 18, 47, 13, 8, 20, 27, 22, 37, 11, 7, 5, 10, 24, 41, 19, 12, 25, 39, 26, 33, 17, 14, 28, 43, 23, 15, 29, 53, 31, 9, 3, 2, 4, 6, 16, 30, 49, 21, 18, 32, 51, 34, 57, 47, 13, 8, 20, 35, 59, 36, 71, 38, 63, 27, 22, 40, 73

Offset

1,2

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 P > 1 not yet present inside another pair of parentheses;

3) always end the content inside a pair of parentheses with the smallest integer R > 1 not yet present inside another pair of parentheses such that the concatenation PR is prime;

4) after a(1) = 1, a(2) = 2, a(3) = 3, 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.

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..11194

Example

Parentheses are added around each pair of terms whose concatenation produces a prime:
1,(2,3),2,(4,7),(5,9),3,2,4,(6,13),(8,11),7,5,(10,19),(12,17),(14,23),(15,31),9,...
Erasing all the parenthesized contents yields
1,(...),2,(...),(...),3,2,4,(....),(....),7,5,(.....),(.....),(.....),(.....),9,...
We see that the remaining terms rebuild the starting sequence.

Crossrefs

Cf. A000040 (the prime numbers), A303950 (remove parentheses with Fibonacci sum).

Sequence in context: A254967 A229012 A207606 * A132439 A338902 A116217

Adjacent sequences: A303842 A303843 A303844 * A303846 A303847 A303848

Keyword

nonn,base,look

Author

Eric Angelini and Jean-Marc Falcoz, May 01 2018

Status

approved