

A177045


The ordering of expressions built from numbers 2 and exponentiations according to their numeric values.


0



1, 2, 3, 4, 7, 8, 9, 5, 6, 17, 18, 21, 22, 23, 15, 16, 19, 20, 49, 50, 51, 59, 60, 63, 64, 65, 43, 44, 45, 46, 47, 48, 57, 58, 61, 62, 149, 150, 153, 154, 155, 181, 182, 183, 191, 192, 195, 196, 197, 136, 137, 138, 139, 140, 141, 147, 148, 151, 152, 175, 176, 177, 178, 179, 180, 189, 190, 193, 194, 478, 479, 480, 488, 489, 492, 493, 494, 578, 579, 582, 583, 584, 610, 611, 612, 620, 621, 624, 625, 626, 12, 13, 14, 132, 133, 134, 135, 439, 440, 441
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OFFSET

1,2


COMMENTS

Let S be the sequence of all possible expressions built from numbers 2 and exponentiations (^), sorted according to their natural structural ordering (2, 2^2, 2^(2^2), (2^2)^2, 2^(2^(2^2)), 2^((2^2)^2) and so on  see the exact definition in Haskell below).
Let S' be S stablesorted according to the numeric values of its elements in ascending order (the stable sorting is a sorting that keeps the order of elements with equal keys  so 2^(2^2) and (2^2)^2 will be kept in the original order).
This sequence is S' where each expression is replaced with its original index (1based) in S; it is a permutation of the natural numbers sequence.


LINKS



PROG

(Haskell) data Expr = Two  Expr :^: Expr
 needed only for presentation
instance Show Expr where show Two = "2"; show (x :^: y) = "(" ++ show x ++ "^" ++ show y ++ ")"
ofSize 1 = [Two]
ofSize n = [left :^: right  k < [1..n1], left < ofSize k, right < ofSize (nk)]
 this defines the S sequence
s = [e  n < [1..], e < ofSize n]


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



