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A094206 a(n) = number of consistent orderings of 1..n based only on factorization. 0

%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

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Last modified November 28 12:20 EST 2022. Contains 358416 sequences. (Running on oeis4.)