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Order the nonnegative integers by increasing binary length of values, then by decreasing binary length of values squared, then by increasing binary length of values cubed, etc.
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%I #11 Mar 01 2023 02:08:59

%S 0,1,3,2,6,7,5,4,12,15,14,13,10,9,8,11,23,24,25,27,30,31,29,28,26,20,

%T 18,17,16,19,21,22,46,47,48,50,49,54,55,59,60,61,63,62,58,56,57,52,51,

%U 53,39,40,36,35,34,32,33,37,38,42,41,43,44,45,91,92,93,94

%N Order the nonnegative integers by increasing binary length of values, then by decreasing binary length of values squared, then by increasing binary length of values cubed, etc.

%C We ignore leading zeros (and 0 is assumed to have binary length 0).

%C This sequence is a permutation of the nonnegative integers with inverse A360983.

%C The order of appearance of two distinct integers, say x and y with x > y, depends on the parity of A360963(x, y): even implies x appears before y, odd implies x appears after y.

%H Rémy Sigrist, <a href="/A360982/b360982.txt">Table of n, a(n) for n = 0..8191</a>

%H Rémy Sigrist, <a href="/A360982/a360982.png">Scatterplot of the first 2^15 terms</a>

%H Rémy Sigrist, <a href="/A360982/a360982.gp.txt">PARI program</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e The first terms, alongside the binary length of their first powers, are:

%e n a(n) w1 w2 w3 w4 w5 w6

%e -- ---- -- -- -- -- -- --

%e 0 0 0

%e 1 1 1

%e 2 3 2 4

%e 3 2 2 3

%e 4 6 3 6 8

%e 5 7 3 6 9

%e 6 5 3 5 7 10

%e 7 4 3 5 7 9

%e 8 12 4 8 11

%e 9 15 4 8 12 16 20 24

%e 10 14 4 8 12 16 20 23

%e 11 13 4 8 12 15

%e 12 10 4 7 10 14

%e 13 9 4 7 10 13 16 20

%e 14 8 4 7 10 13 16 19

%e 15 11 4 7 11

%o (PARI) See Links section.

%Y See A360959 for a similar sequence.

%Y Cf. A360963, A360983 (inverse).

%K nonn,base

%O 0,3

%A _Rémy Sigrist_, Feb 27 2023