%I
%S 1,2,3,5,8,14,21,38,70,129,238,440,504,949,1790,2301,4363,8272,12408,
%T 23604,26675,45724,87781,168549,181989,351076,677339,1306894,1399054,
%U 2709128,2795144,5423805,10525050
%N Number of distinct sums <= 1 of distinct reciprocals of integers <= n.
%e a(3) = 5 counts numbers { 0, 1/3, 1/2, 5/6, 1 }, each of which is can be represented as the sum of distinct reciprocals 1/1, 1/2, and 1/3.
%p s:= proc(n) option remember;
%p `if`(n=0, {0}, map(x> `if`(n1<n*x, x, [x, x+1/n][]), s(n1)))
%p end:
%p a:= n> nops(s(n)):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 23 2012
%t s[_] := s[n] = If[n == 0, {0}, If[n1 < n*#, #, {#, # + 1/n}]& /@ s[n1] // Flatten];
%t a[n_] := Length[s[n]];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 32}] (* _JeanFrançois Alcover_, May 13 2019, after _Alois P. Heinz_ *)
%Y For possibly nondistinct reciprocals, see A212606.
%Y Cf. A020473, A092669, A092671, A208480.
%K nonn
%O 0,2
%A _Max Alekseyev_, May 22 2012
%E a(27)a(32) from _Alois P. Heinz_, May 23 2012
