[seqfan] Counting Certain Finite Sequences

[Leroy Quet]

I just submitted this:

%I A175303
%S A175303 1,2,3,4,5,13,14
%N A175303 a(n) = number of finite sequences using numbers drawn from (1,2,3,...,n), using each number at most once, where each number in a finite sequence is not coprime to the previous number, and each finite sequence must be continued if possible. (See example.) 
%e A175303 For n = 7, we have these 14 finite sequences: 
%e A175303 1; 
%e A175303 2,4,6,3; 
%e A175303 2,6,3; 
%e A175303 2,6,4; 
%e A175303 3,6,2,4; 
%e A175303 3,6,4,2; 
%e A175303 4,2,6,3; 
%e A175303 4,6,2; 
%e A175303 4,6,3; 
%e A175303 5; 
%e A175303 6,2,4; 
%e A175303 6,3; 
%e A175303 6,4,2; 
%e A175303 7 
%K A175303 more,nonn
%O A175303 1,2

Is this clear? Is this idea already named something in the mathematical literature?
(Excuse my ignorance.)

And, is there a formula for directly or recursively calculating the terms of this sequence?

Thanks,
Leroy Quet


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