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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
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 (list; graph; refs; listen; history; text; internal format)
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 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.

LINKS

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

EXAMPLE

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.

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).

Sequence in context: A129708 A071518 A065338 * A294649 A001438 A105587

Adjacent sequences:  A316269 A316270 A316271 * A316273 A316274 A316275

KEYWORD

nonn,base

AUTHOR

Eric Angelini and Jean-Marc Falcoz, Jun 28 2018

STATUS

approved

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Last modified July 21 06:55 EDT 2019. Contains 325192 sequences. (Running on oeis4.)