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A297074
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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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0
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OFFSET
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1,5
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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With[{x = Sqrt@ 2}, Array[Count[#, _?IntegerQ] &@ Map[ToExpression@ StringReplace[ToString@ #, {"{" -> "(", "}" -> ")", ", " -> "^"}] &, Groupings[#, 2] /. _Integer -> x] &, 9]] (* Michael De Vlieger, Dec 24 2017 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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