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A117992 Number of distinct rational numbers less than 1 that can be written as fractions of the first n composite numbers.

%I A117992 #14 Mar 10 2019 08:51:23
%S A117992 0,1,3,5,9,11,17,20,26,29,33,38,50,54,65,80,89,98,102,113,123,144,158,
%T A117992 166,190,203,215,222,236,247,277,287,314,328,346,364,375,400,417,438,
%U A117992 478,487,529,548,573,604,617,641,667,683,698,749,769,796,836,851,873
%N A117992 Number of distinct rational numbers less than 1 that can be written as fractions of the first n composite numbers.
%C A117992 a(n) = (A117991(n) - 1)/2.
%H A117992 Robert Israel, <a href="/A117992/b117992.txt">Table of n, a(n) for n = 1..6000</a>
%e A117992 n=4, the first 4 composite numbers are 4,6, 8 and 9:
%e A117992 a(3)=#{4/6,4/8,4/9,6/8,6/9,8/9}=#{4/9,1/2,2/3,3/4,8/9}=5.
%p A117992 N:= 1000: # to use composites up to N
%p A117992 comps:= remove(isprime, [$4..100]):
%p A117992 nc:= nops(comps):
%p A117992 S:= {}: A[1]:= 0:
%p A117992 for n from 2 to nc do
%p A117992 S:= S union {seq(comps[j]/comps[n],j=1..n-1)};
%p A117992   A[n]:= nops(S);
%p A117992 od:
%p A117992 seq(A[i],i=1..nc); # _Robert Israel_, Jan 30 2018
%t A117992 M = 100; (* to use composites up to M *)
%t A117992 comps = Select[Range[4, M], CompositeQ];
%t A117992 nc = Length[comps]; S = {}; A[1] = 0;
%t A117992 For[n = 2, n <= nc, n++, S = S ~Union~ Table[comps[[j]]/comps[[n]], {j, 1, n - 1}]; A[n] = Length[S]];
%t A117992 Array[A, nc] (* _Jean-François Alcover_, Mar 10 2019, after _Robert Israel_ *)
%Y A117992 Cf. A015614, A117991.
%K A117992 nonn
%O A117992 1,3
%A A117992 _Reinhard Zumkeller_, Apr 08 2006