%I
%S 1,1,1,2,3,5,9,25,66,158,424,1048,2445,5736,17069,88674,241698,648786,
%T 1600339,5379356
%N a(n) = number of consistent orderings of 1..n based only on factorization.
%C Take a set of objects [n] indexed by the positive integers which multiply so that [a] [b] = [ab] (which automatically makes them commute, associate, obey gcd([a],[b])=[gcd(a,b)] etc.) and also partially define a consistent ordering relation < to obey two rules:
%C Rule 1: p<q ==> [p] < [q], for primes p,q and Rule 2: A<B, C<D ==> AC < BD, for any objects A, B, C, D. Rule 2 captures certain intuitive requirements for ordering products  for example specializing A=[1] and C=D captures the idea that "multiples are larger", etc. Sequence gives number of ways of consistently ordering [1]..[n].
%e Up to n=3 there's only one way: [1], [1][2], [1][2][3], but then for n=4=2^2 the rules do not say whether [3]<[4] or [4]<[3], although they do say that [2]<[4], so we get two orderings [1][2][3][4], [1][2][4][3].
%K nonn,nice
%O 1,4
%A _Marc LeBrun_, May 04 2004
