|
| |
|
|
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
|
|
|
|
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).
|
|
|
KEYWORD
|
nonn,base
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
