This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348268 #35 Jan 06 2025 22:06:25 %S A348268 1,2,3,4,5,6,7,8,11,10,9,12,13,14,17,16,19,22,15,20,29,18,21,24,23,26, %T A348268 37,28,31,34,41,32,43,38,33,44,25,30,35,40,53,58,27,36,67,42,51,48,47, %U A348268 46,39,52,61,74,49,56,59,62,73,68,71,82,79,64,83,86,57,76,55,66,77 %N A348268 Mapping between Lyndon factorization and prime factorization. %C A348268 A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. %C A348268 We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1). %C A348268 We use Lyndon factorization on the reversed order binary expansion of n, then we use the mapping from Lyndon words (A328596(k) reversed binary expansion) to positive integers that have no divisors other than 1 and itself (A008578(k+1)). a(n) has factors in A008578 as the binary expansion of n has in A328596. %H A348268 Thomas Scheuerle, <a href="/A348268/b348268.txt">Table of n, a(n) for n = 0..5000</a> %H A348268 Thomas Scheuerle, <a href="/A348268/a348268.m.txt">Program (MATLAB)</a> %H A348268 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a> %F A348268 a(Lyndonproduct(n,m)) = a(n)*a(m). %F A348268 a(1 + 2*n)/a(n) = 2. %F A348268 all a(A329399(n)) are in A000961 (powers of primes). %F A348268 all a(A328595(n)) (reversed binary expansion is a necklace) are in A329131 (prime signature is a Lyndon word). %e A348268 We map Lyndon-words to positive integers that have no divisors other than 1 and itself: [] -> 1, 1 -> 2, 01 -> 3, 001 -> 5, 011 -> 7, 0001 -> 11, ... %e A348268 9 is in reversed order binary: 1001, has the factors (1)(001) -> a(9) = 2*5 = 10. %e A348268 10 is in reversed order binary: 0101, has the factors (01)(01) -> a(10) = 3*3 = 9. %o A348268 (MATLAB) % See Scheuerle link. %Y A348268 Cf. A329313, A328596, A008578, A329399, A000961, A328595, A328595. %K A348268 nonn,base %O A348268 0,2 %A A348268 _Thomas Scheuerle_, Oct 09 2021