%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= and C=D captures the idea that "multiples are larger", etc. Sequence gives number of ways of consistently ordering ..[n].
%e Up to n=3 there's only one way: , , , but then for n=4=2^2 the rules do not say whether < or <, although they do say that <, so we get two orderings , .
%A _Marc LeBrun_, May 04 2004