A307105 - OEIS

%I #78 Mar 02 2020 00:48:43

%S 1,1,3,9,21,63,117,351,621,1161,2043,6129,8631,25893,45135,71685,

%T 102285,306855,420309,1260927,1755513,2671299,4571073,13713219,

%U 17156853,25778169,43930755,59315085,80765235,242295705,295267275,885801825

%N 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 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 a(n) <= 3*a(n-1), with equality iff n is prime. - _Yan Sheng Ang_, Feb 13 2020

%C 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 Yan Sheng Ang, <a href="/A307105/b307105.txt">Table of n, a(n) for n = 0..51</a>

%F a(p) = 3 * a(p-1), for p prime. - _Giovanni Resta_, Jul 07 2019

%e a(2) = 3 because {1,2} can create {1/2, 1, 2}.

%e a(3) = 9 because {1,2,3} can create {1/6, 1/3, 1/2, 2/3, 1, 3/2, 2, 3, 6}.

%e 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 s:= proc(n) option remember; `if`(n=0, {1},

%p       map(x-> [x, x*n, x/n][], s(n-1)))

%p     end:

%p a:= n-> nops(s(n)):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 29 2019

%t L={}; s={1}; Do[s = Union[s, s/k, s*k]; AppendTo[L, Length@ s], {k, 13}]; L (* _Giovanni Resta_, Jul 07 2019 *)

%Y Cf. A018805, A060957.

%K nonn

%O 0,3

%A _Brian Barsotti_, Jul 07 2019

%E a(9)-a(31) from _Giovanni Resta_, Jul 07 2019