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A307105 Number of rational numbers which can be constructed from the set of integers between 1 and n, through a combination of multiplication and division.

%I A307105 #78 Mar 02 2020 00:48:43
%S A307105 1,1,3,9,21,63,117,351,621,1161,2043,6129,8631,25893,45135,71685,
%T A307105 102285,306855,420309,1260927,1755513,2671299,4571073,13713219,
%U A307105 17156853,25778169,43930755,59315085,80765235,242295705,295267275,885801825
%N A307105 Number of rational numbers which can be constructed from the set of integers between 1 and n, through a combination of multiplication and division.
%C A307105 This sequence can contain only odd terms, because apart from 1, for every term x/y there is always the corresponding terms y/x. - _Giovanni Resta_, Jul 07 2019
%C A307105 a(n) <= 3*a(n-1), with equality iff n is prime. - _Yan Sheng Ang_, Feb 13 2020
%C A307105 Conjecture: Let p <= n be prime. If m and p^a*m are two such rationals, then so is p^k*m for all 0 < k < a. - _Yan Sheng Ang_, Feb 13 2020
%H A307105 Yan Sheng Ang, <a href="/A307105/b307105.txt">Table of n, a(n) for n = 0..51</a>
%F A307105 a(p) = 3 * a(p-1), for p prime. - _Giovanni Resta_, Jul 07 2019
%e A307105 a(2) = 3 because {1,2} can create {1/2, 1, 2}.
%e A307105 a(3) = 9 because {1,2,3} can create {1/6, 1/3, 1/2, 2/3, 1, 3/2, 2, 3, 6}.
%e A307105 a(4) = 21 because {1,2,3,4} can create {1/24, 1/12, 1/8, 1/6, 1/4, 1/3, 3/8, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 2, 8/3, 3, 4, 6, 8, 12, 24}.
%p A307105 s:= proc(n) option remember; `if`(n=0, {1},
%p A307105       map(x-> [x, x*n, x/n][], s(n-1)))
%p A307105     end:
%p A307105 a:= n-> nops(s(n)):
%p A307105 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 29 2019
%t A307105 L={}; s={1}; Do[s = Union[s, s/k, s*k]; AppendTo[L, Length@ s], {k, 13}]; L (* _Giovanni Resta_, Jul 07 2019 *)
%Y A307105 Cf. A018805, A060957.
%K A307105 nonn
%O A307105 0,3
%A A307105 _Brian Barsotti_, Jul 07 2019
%E A307105 a(9)-a(31) from _Giovanni Resta_, Jul 07 2019