%I #24 Sep 26 2019 04:18:54
%S 1,2,3,3,4,4,5,6,6,7,8,8,9,9,10,11,11,12,13,13,14,14,15,16,16,17,17,
%T 18,18,19,20,20,21,22,22,23,23,24,25,25
%N Maximum sum of base lengths over all minimal factorizations of lengthn binary strings.
%C A factorization of a binary string x is an expression of the form prod_i w_i^{e_i}, where each w_i is a word and e_i is an integer exponent specifying how many times the word is repeated. For example 0101000 = (01)^2 0^3. A minimal factorization is one that minimizes the weight of the factorization, which is defined to be sum of the lengths of the w_i. a(n) then measures the maximum weight over all lengthn binary strings.
%C Since there are arbitrarily long binary words having no repetitions larger than squares (Thue 1906), we see that a(n) >= n/2. By considering a(14) = 9, and splitting a word into blocks of size 14 and one left over, we see that a(n) <= 0.644 n for sufficiently large n.
%C Upper bound for sufficiently large n reduced to a(n) < 0.621 n considering a(29) = 18.  _Bert Dobbelaere_, Jul 21 2019
%F a(j+k) <= a(j) + a(k).  _Bert Dobbelaere_, Jul 21 2019
%e For n = 8, we have a(8) = 6, and a word that achieves the maximum is 01001101, where the corresponding weight6 factorization is (01) 0^2 1^2 (01).
%Y Cf. A309078.
%K nonn,more
%O 1,2
%A _Jeffrey Shallit_, Jul 11 2019
%E a(21)a(40) from _Bert Dobbelaere_, Jul 21 2019
