A333777 -id:A333777 - cp's OEIS frontend

A333776 Scan the binary representation of n from right to left; at each 1, reverse the bits to the right and excluding this 1. The resulting binary representation is that of a(n).

0, 1, 2, 3, 4, 6, 5, 7, 8, 12, 10, 14, 9, 11, 13, 15, 16, 24, 20, 28, 18, 22, 26, 30, 17, 19, 21, 23, 25, 29, 27, 31, 32, 48, 40, 56, 36, 44, 52, 60, 34, 38, 42, 46, 50, 58, 54, 62, 33, 35, 37, 39, 41, 45, 43, 47, 49, 57, 53, 61, 51, 55, 59, 63, 64, 96, 80

Offset: 0

Rémy Sigrist, Apr 05 2020

This sequence is a permutation of the nonnegative integers (as it is injective and preserves the binary length); see A333777 for the inverse.

We can devise a variant of this sequence for any fixed base b > 1, by performing a reversal at each nonzero digit in base b.

			For n = 90:
- the binary representation of 90 is "1011010",
- this binary representation evolves as follows (parentheses indicate reversals):
    1 0 1 1 0 1(0)
    1 0 1 1(0 1 0)
    1 0 1(0 1 0 1)
    1(1 0 1 0 1 0)
- the resulting binary representation is "1101010"
- and a(90) = 106.
The binary plot of the first terms is as follows (#'s denote 1's):
                                  ################################
                  ################ # #  ##    ####        ########
          ######## # #  ##    ####  ## # #  ## # #    #### # #  ##
      #### # #  ##  ## # #  ## # #    #### # #  ##  ## # #  ## # #
    ## # #  ## # #    #### # #  ##        ######## # #  ##    ####
   # #  ##    ####        ########                ################
            1         2         3         4         5         6
  0123456789012345678901234567890123456789012345678901234567890123

See A333692 for a similar sequence.

Cf. A000120, A330081, A333777 (inverse), A333778 (fixed points).

a(n, base=2) = { my (d=digits(n, base), t=[]); forstep (k=#d, 1, -1, if (d[k], t=Vecrev(t)); t=concat(d[k], t)); fromdigits(t, base); }

a(2*n) <= 2*a(n) with equality iff n = 0 or n is a power of 2.

A000120(a(n)) = A000120(n).

A367307 Reverse bits in blocks in binary expansion of n where blocks are separated by every second 1 bit starting from the most significant 1 bit as the first separator.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 12, 10, 13, 9, 11, 14, 15, 16, 24, 20, 25, 18, 21, 26, 27, 17, 19, 22, 23, 28, 30, 29, 31, 32, 48, 40, 49, 36, 41, 50, 51, 34, 37, 42, 43, 52, 54, 53, 55, 33, 35, 38, 39, 44, 46, 45, 47, 56, 60, 58, 61, 57, 59, 62, 63, 64, 96, 80, 97

Offset: 0

Mikhail Kurkov, Nov 13 2023

Self-inverse permutation of nonnegative integers.

			For n = 100930, reversals are each (...) block
  n    = 100930 = binary 1(1000)1(0100)1(000010)
  a(n) = 72016  = binary 1(0001)1(0010)1(010000)

Cf. A053644, A053645, A059893, A333776, A333777.

a(n) = my(v1); v1 = binary(n); my(A = 1); while(A <= #v1, my(B = A, C = 0, D); A++; while(A <= #v1, C += v1[A]; if(v1[A] && (C == 1), D = A); if(C < 2, A++, break)); if(C > 0, v1[D] = 0; v1[A + B - D] = 1)); fromdigits(v1, 2)

a(n) = b(c(f(n)) + g(n)) = b(h(c(n))) for n > 0 with a(0) = 0 where
b(n) = b(f(h(n))) + g(n) for n > 0 with b(0) = 0,
c(n) = h(c(f(n)) + g(n)) for n > 0 with c(0) = 0,
f(n) = A053645(n), g(n) = A053644(n) and where h(n) = A059893(n).
Note that c (A333776) is the inverse permutation to b (A333777).
a(n) < 2^k iff n < 2^k for k >= 0.

A333778 Fixed points of A333776.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 15, 16, 29, 31, 32, 36, 42, 57, 63, 64, 86, 113, 127, 128, 136, 170, 225, 251, 255, 256, 292, 338, 449, 477, 499, 511, 512, 528, 588, 674, 682, 897, 949, 995, 1023, 1024, 1172, 1346, 1390, 1793, 1849, 1893, 1987, 2039, 2047, 2048, 2080

Offset: 1

Rémy Sigrist, Apr 05 2020

This sequence contains A000079, A000225.

Cf. A000079, A000225, A333776, A333777.

is(n, base=2) = { my (d=digits(n, base), t=[]); forstep (k=#d, 1, -1, if (d[k], t=Vecrev(t);); t=concat(d[k], t)); n==fromdigits(t, base) }

cp's OEIS Frontend

A333776 Scan the binary representation of n from right to left; at each 1, reverse the bits to the right and excluding this 1. The resulting binary representation is that of a(n).

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A367307 Reverse bits in blocks in binary expansion of n where blocks are separated by every second 1 bit starting from the most significant 1 bit as the first separator.

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A333778 Fixed points of A333776.

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