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 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). 5
 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, 18, 28, 20, 24, 16, 63, 47, 55, 39, 59, 43, 45, 35, 61, 46, 51, 37, 53, 38, 41, 33, 62, 54, 57, 42, 58, 44, 49, 34, 60, 50, 52, 36, 56, 40, 48, 32, 127, 95, 111, 79 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS In other words, for any n > 0, A000120(n) = 1 + A023416(a(n)). This sequence is a self-inverse permutation of the natural numbers, with fixed points A031448. We can build an analog of this sequence for any base b > 1: - let s_b be the sum of digits in base b, - in particular s_2 = A000120 and s_10 = A007953, - let l_b be the number of digits in base b, - in particular l_2 = A070939 and l_10 = A055642, - 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)), - in particular, f_2 = a (this sequence), - f_b is a self-inverse permutation of the natural numbers, - l_b(n) = l_b(f_b(n)) for any n > 0, - f_b(b^k) = b^(k+1) - 1 for any k >= 0, - see also scatterplots of f_3 and f_10 in Links section. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..8191 Rémy Sigrist, PARI program for A298847 Rémy Sigrist, Colored scatterplot of the first 2^16 - 1 terms (where the color is function of the Hamming weight of n) Rémy Sigrist, Scatterplot of the first 3^9 - 1 terms of f_3 Rémy Sigrist, Scatterplot of the first 10^4 - 1 terms of f_10 FORMULA A070939(n) = A070939(a(n)) for any n > 0. a(2^k) = 2^(k+1) - 1 for any k >= 0. A000120(n) + A000120(a(n)) = 1 + A070939(n) for any n > 0. EXAMPLE The first terms, alongside the binary representations of n and of a(n), are:   n     a(n)    bin(n)    bin(a(n))   --    ----    ------    ---------    1       1         1         1    2       3        10        11    3       2        11        10    4       7       100       111    5       5       101       101    6       6       110       110    7       4       111       100    8      15      1000      1111    9      11      1001      1011   10      13      1010      1101   11       9      1011      1001   12      14      1100      1110   13      10      1101      1010   14      12      1110      1100   15       8      1111      1000 PROG (PARI) See Links section. CROSSREFS Cf. A000120, A007953, A023416, A031448, A055642, A070939. Sequence in context: A303075 A286417 A303076 * A059894 A307544 A126314 Adjacent sequences:  A298844 A298845 A298846 * A298848 A298849 A298850 KEYWORD nonn,base AUTHOR Rémy Sigrist, Jan 27 2018 STATUS approved

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Last modified June 30 03:34 EDT 2022. Contains 354913 sequences. (Running on oeis4.)