A375375 -id:A375375 - cp's OEIS frontend

A375374 Let X be the sequence of power towers built of 2's and 3's, sorted first by their height and then colexicographically: 2, 3, 2^2, 3^2, 2^3, 3^3, 2^2^2, 3^2^2, etc. Sequence gives the permutation of indices which reorders X by magnitude.

1, 2, 3, 5, 4, 7, 6, 8, 11, 9, 12, 10, 15, 16, 13, 14, 17, 18, 23, 24, 19, 20, 25, 26, 21, 22, 31, 32, 33, 34, 27, 28, 29, 30, 35, 36, 37, 38, 47, 48, 49, 50, 39, 40, 41, 42, 51, 52, 53, 54, 43, 44, 45, 46, 63, 64, 65, 66, 67, 68, 69, 70, 55, 56, 57, 58, 59, 60

Offset: 1

Pontus von Brömssen, Aug 13 2024

nonn

The terms are less dispersed here compared to A185969, because colex order is more correlated to the magnitude of the power tower than lex order is, i.e., we often get a smaller value of the power tower by putting the small numbers high up in the tower. Specifically, the only integers x, y >= 2 for which x < y and x^y < y^x is x = 2, y = 3.

Index entries for sequences that are permutations of the natural numbers.

3rd row of A375376.

Cf. A081241, A185969 (lexicographic instead of colexicographic order), A375375 (inverse permutation).

a(2*n-1) = 2*a(n-1)+1 and a(2*n) = a(2*n-1)+1 for n >= 7.

a(n) = A081241(A185969(n)).

A375377 Square array read by antidiagonals: the n-th row is the inverse to the permutation given by the n-th row of A375376.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 5, 3, 2, 1, 7, 6, 4, 4, 3, 2, 1, 8, 7, 7, 5, 4, 3, 2, 1, 9, 8, 6, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 7, 5, 4, 3, 3, 1, 11, 10, 10, 8, 6, 6, 6, 4, 2, 2, 1, 12, 11, 12, 9, 8, 7, 7, 5, 5, 3, 2, 1, 13, 12, 9, 10, 9, 8, 5, 6, 6, 4, 4, 2, 1

Offset: 1

Pontus von Brömssen, Aug 14 2024

			Array begins:
   n=1: 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=2: 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=3: 1, 2, 3, 5, 4, 7, 6,  8, 10, 12,  9, 11, 15, 16, 13, ...
   n=4: 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=5: 1, 2, 3, 4, 5, 7, 6,  8,  9, 12, 10, 13, 15, 17, 11, ...
   n=6: 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=7: 1, 2, 3, 4, 6, 7, 5, 10, 11,  8, 12, 14,  9, 13, 15, ...
   n=8: 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=9: 1, 3, 2, 5, 6, 8, 4,  7, 10, 13, 11, 14, 17, 18,  9, ...
  n=10: 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, ...
  n=11: 1, 2, 4, 3, 6, 8, 5,  9, 12, 10, 13, 17,  7, 11, 16, ...
  n=12: 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, ...
  n=13: 1, 2, 4, 3, 5, 8, 6, 10, 12,  9, 14, 15,  7, 11, 13, ...
  n=14: 1, 2, 3, 4, 5, 7, 6,  9, 10,  8, 11, 12, 13, 14, 15, ...
  n=15: 1, 2, 3, 5, 4, 7, 8, 11,  6, 12, 14, 17,  9, 15, 19, ...
For n = 7 = 2^0 + 2^1 + 2^2, the set S (defined in A375376) is {0+2, 1+2, 2+2} = {2, 3, 4}. The first power towers formed by 2's, 3's, and 4's, in colex order, together with their ranks (by magnitude) are:
   k | power tower | rank T(7,k)
   --+-------------+------------
   1 |     2 = 2   |     1
   2 |     3 = 3   |     2
   3 |     4 = 4   |     3
   4 |   2^2 = 4   |     4
   5 |   3^2 = 9   |     6
   6 |   4^2 = 16  |     7
   7 |   2^3 = 8   |     5
   8 |   3^3 = 27  |    10
   9 |   4^3 = 64  |    11
  10 |   2^4 = 16  |     8
  11 |   3^4 = 81  |    12
  12 |   4^4 = 256 |    14
  13 | 2^2^2 = 16  |     9
  14 | 3^2^2 = 81  |    13
  15 | 4^2^2 = 256 |    15

Index entries for sequences that are permutations of the natural numbers.

Cf. A375375 (3rd row), A375376 (the inverse permutation to each row).

cp's OEIS Frontend

A375374 Let X be the sequence of power towers built of 2's and 3's, sorted first by their height and then colexicographically: 2, 3, 2^2, 3^2, 2^3, 3^3, 2^2^2, 3^2^2, etc. Sequence gives the permutation of indices which reorders X by magnitude.

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