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A381839 In the binary expansion of n (without leading zeros): complement the bits strictly between the leftmost and the rightmost 0's, if any.

%I A381839 #14 Mar 10 2025 11:11:50
%S A381839 0,1,2,3,4,5,6,7,10,9,8,11,12,13,14,15,22,21,20,19,18,17,16,23,26,25,
%T A381839 24,27,28,29,30,31,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,47,54,
%U A381839 53,52,51,50,49,48,55,58,57,56,59,60,61,62,63,94,93,92,91
%N A381839 In the binary expansion of n (without leading zeros): complement the bits strictly between the leftmost and the rightmost 0's, if any.
%C A381839 This sequence is a self-inverse permutation of the nonnegative integers.
%C A381839 This sequence has similarities with A122155 (where we complement bits between the leftmost and the rightmost 1's).
%C A381839 This sequence has infinitely many fixed points: A000225, A030130, and the positive numbers whose binary expansion have exactly two 0's that are also adjacent to each other.
%H A381839 Rémy Sigrist, <a href="/A381839/b381839.txt">Table of n, a(n) for n = 0..8191</a>
%H A381839 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A381839 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A381839 a(2*n + 1) = 2*a(n) + 1.
%e A381839 The first terms, in decimal and in binary, are:
%e A381839   n   a(n)  bin(n)  bin(a(n))
%e A381839   --  ----  ------  ---------
%e A381839    0     0       0          0
%e A381839    1     1       1          1
%e A381839    2     2      10         10
%e A381839    3     3      11         11
%e A381839    4     4     100        100
%e A381839    5     5     101        101
%e A381839    6     6     110        110
%e A381839    7     7     111        111
%e A381839    8    10    1000       1010
%e A381839    9     9    1001       1001
%e A381839   10     8    1010       1000
%e A381839   11    11    1011       1011
%e A381839   12    12    1100       1100
%e A381839   13    13    1101       1101
%e A381839   14    14    1110       1110
%e A381839   15    15    1111       1111
%e A381839   16    22   10000      10110
%o A381839 (PARI) a(n) = { my (b = binary(n)); for (i = 1, #b, if (b[i]==0, forstep (j = #b, 1, -1, if (b[j]==0, for (k = i+1, j-1, b[k] = 1-b[k];); return (fromdigits(b, 2)););););); return (n); }
%o A381839 (Python)
%o A381839 def a(n):
%o A381839     b = bin(n)[2:]
%o A381839     zl, zr = b.find('0'), b.rfind('0')
%o A381839     return n if abs(zl-zr) < 2 else int(b[:zl+1]+"".join('0' if bi == '1' else '1' for bi in b[zl+1:zr])+b[zr:], 2)
%o A381839 print([a(n) for n in range(70)]) # _Michael S. Branicky_, Mar 09 2025
%o A381839 (Python)
%o A381839 def A381839(n): return n^((1<<n.bit_length()-t-1)-1)^(((~n&n+1)<<1)-1) if n and (t:=(s:=bin(n)[2:]).find('0'))!=-1 and s.count('0')>1 else n # _Chai Wah Wu_, Mar 09 2025
%Y A381839 Cf. A000225, A030130, A122155, A381852.
%K A381839 nonn,base,easy
%O A381839 0,3
%A A381839 _Rémy Sigrist_, Mar 08 2025