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A373886 a(n) is the least number whose binary expansion can be obtained by reversing one or more consecutive bits in the binary expansion of n.

%I A373886 #57 Aug 14 2024 11:06:06
%S A373886 0,1,1,3,1,3,3,7,1,3,5,7,3,7,7,15,1,3,6,7,5,11,13,15,3,7,11,15,7,15,
%T A373886 15,31,1,3,6,7,9,13,14,15,5,11,21,23,13,27,29,31,3,7,14,15,11,23,27,
%U A373886 31,7,15,23,31,15,31,31,63,1,3,6,7,12,13,14,15,9,19
%N A373886 a(n) is the least number whose binary expansion can be obtained by reversing one or more consecutive bits in the binary expansion of n.
%C A373886 This sequence has similarities with A087734; here we reverse some consecutive bits, there we negate some consecutive bits.
%H A373886 Rémy Sigrist, <a href="/A373886/b373886.txt">Table of n, a(n) for n = 0..8190</a>
%H A373886 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A373886 a(n) <= n with equality iff n belongs to A000225.
%F A373886 a^k(n) = A038573(n) for k sufficiently large (where a^k denotes the k-th iterate of the sequence).
%F A373886 a(2^k) = 1 for any k >= 0.
%F A373886 A000120(a(n)) = A000120(n).
%e A373886 The first terms, alongside their binary expansion, are:
%e A373886   n   a(n)  bin(n)  bin(a(n))
%e A373886   --  ----  ------  ---------
%e A373886    0     0       0          0
%e A373886    1     1       1          1
%e A373886    2     1      10          1
%e A373886    3     3      11         11
%e A373886    4     1     100          1
%e A373886    5     3     101         11
%e A373886    6     3     110         11
%e A373886    7     7     111        111
%e A373886    8     1    1000          1
%e A373886    9     3    1001         11
%e A373886   10     5    1010        101
%e A373886   11     7    1011        111
%e A373886   12     3    1100         11
%e A373886   13     7    1101        111
%e A373886   14     7    1110        111
%e A373886   15    15    1111       1111
%e A373886   16     1   10000          1
%p A373886 f:= proc(n) local L,nL,i,j,k,r,x;
%p A373886   L:= convert(n,base,2);
%p A373886   nL:= nops(L);
%p A373886   r:= n;
%p A373886   for i from 1 to nL-1 do
%p A373886     for j from i+1 to nL do
%p A373886       r:= min(r, n + add((L[j-k]-L[i+k])*2^(i+k-1),k=0..j-i));
%p A373886   od od;
%p A373886   r
%p A373886 end proc:
%p A373886 map(f, [$0..100]); # _Robert Israel_, Aug 13 2024
%o A373886 (PARI) a(n, base = 2) = { my (d = if (n, digits(n, base), [0])); setbinop((i, j) -> fromdigits(concat([d[1..i-1], Vecrev(d[i..j]), d[j+1..#d]]), base), [1..#d])[1]; }
%o A373886 (Python)
%o A373886 def a(n):
%o A373886     b = bin(n)[2:]
%o A373886     return min(int(b[:i]+b[i:j][::-1]+b[j:], 2) for i in range(len(b)) for j in range(i, len(b)+1))
%o A373886 print([a(n) for n in range(74)]) # _Michael S. Branicky_, Aug 13 2024
%Y A373886 Cf. A000120, A000225, A038573, A087734, A331856.
%K A373886 nonn,base,look
%O A373886 0,4
%A A373886 _Rémy Sigrist_, Aug 10 2024