A375374 -id:A375374 - cp's OEIS frontend

A375375 Inverse permutation to A375374.

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

Offset: 1

Pontus von Brömssen, Aug 13 2024

nonn

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

3rd row of A375377.

Cf. A081241, A256231, A375374 (inverse permutation).

a(n) = A256231(A081241(n)).

A185969 Let S be the sequence of power towers built of 2 and 3 sorted by their height and for equal heights - in lexicographic order: 2, 3, 2^2, 2^3, 3^2, 3^3, 2^2^2, 2^2^3 etc. A(n) = the permutation of indexes which reorders S by magnitude.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 8, 9, 12, 13, 15, 23, 10, 14, 19, 27, 16, 24, 17, 25, 20, 28, 21, 29, 31, 47, 39, 55, 18, 26, 22, 30, 35, 51, 43, 59, 32, 48, 40, 56, 33, 49, 41, 57, 36, 52, 44, 60, 37, 53, 45, 61, 63, 95, 79, 111, 71, 103, 87, 119, 34, 50, 42, 58, 38

Offset: 1

Vladimir Reshetnikov, Feb 07 2011

			a(6) =  7; tower(7)  = 2^2^2 = 2^4 =  16.
a(7) =  6; tower(6)  = 3^3   =        27.
a(8) = 11; tower(11) = 3^2^2 = 3^4 =  81.
a(9) =  8; tower(8)  = 2^2^3 = 2^8 = 256.

Alois P. Heinz, Table of n, a(n) for n = 1..14335
Vladimir Reshetnikov, 2-3 sequence puzzle, SeqFan list, Mar 18 2015
Index entries for sequences that are permutations of the natural numbers

Cf. A032810, A081241, A248907, A256179, A256231, A375374 (colexicographic instead of lexicographic order).

a(2*n-1) = A081241(2*A081241(a(n-1))+1) and a(2*n) = A081241(A081241(a(2*n-1))+1) for n >= 7. - Pontus von Brömssen, Aug 10 2024

More terms from Alois P. Heinz, Apr 05 2011

A375376 Square array read by antidiagonals: Let n = Sum_{i=1..m} 2^e_i be the binary expansion of n, let S be the set {e_i+2; 1 <= i <= m}, and let X be the sequence of power towers built of numbers in S, sorted first by their height and then colexicographically. The n-th row of the array gives the permutation of indices which reorders X by magnitude. In case of ties, keep the colexicographic order.

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

Offset: 1

Pontus von Brömssen, Aug 14 2024

Each row is a permutation of the positive integers.

If n is a power of 2, the set S contains a single number and the n-th row is the identity permutation.

			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, 11,  9, 12, 10, 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, 11, 15, 10, 12, 16, 13, ...
   n=6: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=7: 1, 2, 3, 4, 7, 5,  6, 10, 13,  8,  9, 11, 14, 12, 15, ...
   n=8: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
   n=9: 1, 3, 2, 7, 4, 5,  8,  6, 15,  9, 11, 16, 10, 12, 17, ...
  n=10: 1, 2, 3, 4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
  n=11: 1, 2, 4, 3, 7, 5, 13,  6,  8, 10, 14,  9, 11, 22, 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, 7, 13,  6, 10,  8, 14,  9, 15, 11, 12, ...
  n=14: 1, 2, 3, 4, 5, 7,  6, 10,  8,  9, 11, 12, 13, 14, 15, ...
  n=15: 1, 2, 3, 5, 4, 9,  6,  7, 13, 21,  8, 10, 17, 11, 14, ...
For n = 7 = 2^0 + 2^1 + 2^2, the set S is {0+2, 1+2, 2+2} = {2, 3, 4}. The smallest power towers formed by 2's, 3's, and 4's, together with their colex ranks are:
   k | power tower | colex rank T(7,k)
   --+-------------+------------------
   1 |     2 = 2   |        1
   2 |     3 = 3   |        2
   3 |     4 = 4   |        3
   4 |   2^2 = 4   |        4
   5 |   2^3 = 8   |        7
   6 |   3^2 = 9   |        5
   7 |   4^2 = 16  |        6
   8 |   2^4 = 16  |       10
   9 | 2^2^2 = 16  |       13
  10 |   3^3 = 27  |        8
  11 |   4^3 = 64  |        9
  12 |   3^4 = 81  |       11
  13 | 3^2^2 = 81  |       14
  14 |   4^4 = 256 |       12
  15 | 4^2^2 = 256 |       15

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

Cf. A185969, A299229, A375374 (3rd row), A375377 (the inverse permutation to each row).

cp's OEIS Frontend

A375375 Inverse permutation to A375374.

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A185969 Let S be the sequence of power towers built of 2 and 3 sorted by their height and for equal heights - in lexicographic order: 2, 3, 2^2, 2^3, 3^2, 3^3, 2^2^2, 2^2^3 etc. A(n) = the permutation of indexes which reorders S by magnitude.

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