OFFSET
1,3
COMMENTS
a(n) = 2*a(n-1) if n is an odd prime, because (p/q)/n and p/(q/n)=(p/q)*n give exactly two different values for each of the different values p/q from the parenthesizations of 1/.../n-1 and a(n) <= 2*a(n-1) if n is not a prime. - Alois P. Heinz, Nov 23 2008
Let M(n) be the smallest integer among the a(n) values. It seems that, for n >= 4, M(n) = A055204, the squarefree part of n!. - Giovanni Resta, Dec 16 2012
EXAMPLE
For n=4, ((1/2)/3)/4 = 1/24, (1/2)/(3/4) = 2/3, (1/(2/3))/4 = 3/8, 1/((2/3)/4) = 6 and 1/(2/(3/4)) = 3/8, giving 4 different values 1/24, 3/8, 2/3 and 6. Thus a(4) = 4.
a(5) = 2*a(4) = 2*4 = 8, because 5 is a prime; the 8 different values are: 1/120, 3/40, 2/15, 5/24, 6/5, 15/8, 10/3, 30. - Alois P. Heinz, Nov 23 2008
MAPLE
p:= proc(n) option remember; local x;
if n<1 then {}
elif n=1 then {1}
elif n=2 then {1/2}
else {seq([x/n, x*n][], x=p(n-1))}
fi
end:
a:= n-> nops(p(n)):
seq(a(n), n=1..20); # Alois P. Heinz, Nov 23 2008
MATHEMATICA
p[0] = {}; p[1] = {1}; p[2] = {1/2}; p[n_] := p[n] = Union[ Flatten[ Table[ {x/n, x*n}, {x, p[n - 1]}]]]; a[n_] := Length[p[n]]; A078389 = Table[an = a[n]; Print[an]; an, {n, 1, 30}] (* Jean-François Alcover, Jan 06 2012, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
John W. Layman, May 07 2003
EXTENSIONS
Corrected a(5)-a(10) and extended a(11)-a(31) by Alois P. Heinz, Nov 23 2008
a(32)-a(37) from Alois P. Heinz, Mar 07 2011
STATUS
approved