cp's OEIS Frontend

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.

%I A316272 #13 Jul 01 2018 03:56:36
%S A316272 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,
%T A316272 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,
%U A316272 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
%N 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.
%C A316272 The sequence is fractal-like as it embeds an infinite number of copies of itself.
%C A316272 The sequence was built according to these rules (see, in the Example section, the parenthesization technique):
%C A316272   1) no overlapping pairs of parentheses;
%C A316272   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 ;
%C A316272   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;
%C A316272   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.
%H A316272 Eric Angelini, <a href="/A316272/b316272.txt">Table of n, a(n) for n = 1..20706</a>
%e A316272 Parentheses are added around each pair of terms made of a composite and a prime number (in any order):
%e A316272 (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,...
%e A316272 Erasing all the parenthesized contents yields
%e A316272 (...),(...),1,(...),2,3,(...),4,1,6,(....),5,2,3,7,(.....),8,4,1,6,9,(.....),11,...
%e A316272 We see that the remaining terms rebuild the starting sequence.
%Y A316272 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).
%K A316272 nonn,base
%O A316272 1,2
%A A316272 _Eric Angelini_ and _Jean-Marc Falcoz_, Jun 28 2018