A078389 Number of different values obtained by evaluating all different parenthesizations of 1/2/3/4/.../n.

1, 1, 2, 4, 8, 16, 32, 60, 116, 192, 384, 544, 1088, 1736, 2576, 3824, 7648, 10352, 20704, 28096, 40256, 62128, 124256, 155488, 227872, 349248, 470352, 622128, 1244256, 1499232, 2998464, 3796224, 5289920, 8048544, 10668096, 12562752, 25125504

Offset: 1

Author

John W. Layman, May 07 2003

Keywords

• nonn
• nice

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

Examples

			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
		

Links

• Index entries for sequences related to parenthesizing

Programs

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 *)

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