A078389 - OEIS

%I #31 Oct 08 2017 18:12:43

%S 1,1,2,4,8,16,32,60,116,192,384,544,1088,1736,2576,3824,7648,10352,

%T 20704,28096,40256,62128,124256,155488,227872,349248,470352,622128,

%U 1244256,1499232,2998464,3796224,5289920,8048544,10668096,12562752,25125504

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

%C 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

%C 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

%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>

%e 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.

%e 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

%p p:= proc(n) option remember; local x;

%p       if n<1 then {}

%p     elif n=1 then {1}

%p     elif n=2 then {1/2}

%p     else {seq([x/n, x*n][], x=p(n-1))}

%p       fi

%p     end:

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

%p seq(a(n), n=1..20);  # _Alois P. Heinz_, Nov 23 2008

%t 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_ *)

%K nonn,nice

%O 1,3

%A _John W. Layman_, May 07 2003

%E Corrected a(5)-a(10) and extended a(11)-a(31) by _Alois P. Heinz_, Nov 23 2008

%E a(32)-a(37) from _Alois P. Heinz_, Mar 07 2011