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

0, 0, 1, 1, 2, 5, 10, 23, 55

Offset: 1

Author

Jon E. Schoenfield, Dec 24 2017

Keywords

• nonn
• more

Comments

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.

Examples

			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.
		

Links

• 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

Crossrefs

Cf. A002845, A055113, A082499, A198683.

Programs

Mathematica

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