A082499 -id:A082499 - cp's OEIS frontend

The largest value that can be obtained by inserting parentheses in x^x^x^x^x^x^x^x^x (9 x's), where x = sqrt(2), is x^(x^((((((x^x)^x)^x)^x)^x)^x)) = 2^128 = 340282366920938463463374607431768211456; this is one of the a(9) = 55 ways of inserting parentheses in x^x^x^x^x^x^x^x^x that yield an integer value.

			With x = sqrt(2),
x = 1.414213... is not an integer, so a(1) = 0;
x^x = 1.632526... is not an integer, so a(2) = 0.
(x^x)^x = 2 is an integer, but x^(x^x) = 1.760839... is not, so a(3) = 1;
((x^x)^x)^x, (x^x)^(x^x), (x^(x^x))^x, and x^(x^(x^x)) are noninteger values, but x^((x^x)^x) = 2, so a(4) = 1;
the only ways of inserting parentheses in x^x^x^x^x that yield integer values are x^(x^((x^x)^x)) = 2 and (((x^x)^x)^x)^x = 4, so a(5) = 2.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy, with permission)
Index entries for sequences related to parenthesizing

Cf. A002845, A055113, A082499, A198683.

With[{x = Sqrt@ 2}, Array[Count[#, ?IntegerQ] &@ Map[ToExpression@ StringReplace[ToString@ #, {"{" -> "(", "}" -> ")", "," -> "^"}] &, Groupings[#, 2] /. _Integer -> x] &, 9]] (* _Michael De Vlieger, Dec 24 2017 *)

cp's OEIS Frontend

A082543 Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible ways. Sequence gives number of distinct integer values when x=sqrt(2).

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A003008 Number of n-level ladder expressions with A030798.

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A297074 Number of ways of inserting parentheses in x^x^...^x (with n x's) whose result is an integer where x = sqrt(2).

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Mathematica