cp's OEIS Frontend

A298847 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the number of ones in the binary expansion of n equals one plus the number of zeros in the binary expansion of a(n).

%I A298847 #17 Jan 28 2018 13:34:36
%S A298847 1,3,2,7,5,6,4,15,11,13,9,14,10,12,8,31,23,27,19,29,21,22,17,30,25,26,
%T A298847 18,28,20,24,16,63,47,55,39,59,43,45,35,61,46,51,37,53,38,41,33,62,54,
%U A298847 57,42,58,44,49,34,60,50,52,36,56,40,48,32,127,95,111,79
%N A298847 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the number of ones in the binary expansion of n equals one plus the number of zeros in the binary expansion of a(n).
%C A298847 In other words, for any n > 0, A000120(n) = 1 + A023416(a(n)).
%C A298847 This sequence is a self-inverse permutation of the natural numbers, with fixed points A031448.
%C A298847 We can build an analog of this sequence for any base b > 1:
%C A298847 - let s_b be the sum of digits in base b,
%C A298847 - in particular s_2 = A000120 and s_10 = A007953,
%C A298847 - let l_b be the number of digits in base b,
%C A298847 - in particular l_2 = A070939 and l_10 = A055642,
%C A298847 - let f_b be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, s_b(n) = 1 + (b-1) * l_b(a(n)) - s_b(a(n)),
%C A298847 - in particular, f_2 = a (this sequence),
%C A298847 - f_b is a self-inverse permutation of the natural numbers,
%C A298847 - l_b(n) = l_b(f_b(n)) for any n > 0,
%C A298847 - f_b(b^k) = b^(k+1) - 1 for any k >= 0,
%C A298847 - see also scatterplots of f_3 and f_10 in Links section.
%H A298847 Rémy Sigrist, <a href="/A298847/b298847.txt">Table of n, a(n) for n = 1..8191</a>
%H A298847 Rémy Sigrist, <a href="/A298847/a298847.gp.txt">PARI program for A298847</a>
%H A298847 Rémy Sigrist, <a href="/A298847/a298847.png">Colored scatterplot of the first 2^16 - 1 terms</a> (where the color is function of the Hamming weight of n)
%H A298847 Rémy Sigrist, <a href="/A298847/a298847_1.png">Scatterplot of the first 3^9 - 1 terms of f_3</a>
%H A298847 Rémy Sigrist, <a href="/A298847/a298847_2.png">Scatterplot of the first 10^4 - 1 terms of f_10</a>
%H A298847 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A298847 A070939(n) = A070939(a(n)) for any n > 0.
%F A298847 a(2^k) = 2^(k+1) - 1 for any k >= 0.
%F A298847 A000120(n) + A000120(a(n)) = 1 + A070939(n) for any n > 0.
%e A298847 The first terms, alongside the binary representations of n and of a(n), are:
%e A298847   n     a(n)    bin(n)    bin(a(n))
%e A298847   --    ----    ------    ---------
%e A298847    1       1         1         1
%e A298847    2       3        10        11
%e A298847    3       2        11        10
%e A298847    4       7       100       111
%e A298847    5       5       101       101
%e A298847    6       6       110       110
%e A298847    7       4       111       100
%e A298847    8      15      1000      1111
%e A298847    9      11      1001      1011
%e A298847   10      13      1010      1101
%e A298847   11       9      1011      1001
%e A298847   12      14      1100      1110
%e A298847   13      10      1101      1010
%e A298847   14      12      1110      1100
%e A298847   15       8      1111      1000
%o A298847 (PARI) See Links section.
%Y A298847 Cf. A000120, A007953, A023416, A031448, A055642, A070939.
%K A298847 nonn,base
%O A298847 1,2
%A A298847 _Rémy Sigrist_, Jan 27 2018