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A094206 a(n) = number of consistent orderings of 1..n based only on factorization. 0
1, 1, 1, 2, 3, 5, 9, 25, 66, 158, 424, 1048, 2445, 5736, 17069, 88674, 241698, 648786, 1600339, 5379356 (list; graph; refs; listen; history; text; internal format)



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:

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].


Table of n, a(n) for n=1..20.


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].


Sequence in context: A101581 A287915 A105180 * A278119 A118998 A276410

Adjacent sequences:  A094203 A094204 A094205 * A094207 A094208 A094209




Marc LeBrun, May 04 2004



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Last modified January 19 06:18 EST 2021. Contains 340266 sequences. (Running on oeis4.)