A333693 -id:A333693 - cp's OEIS frontend

A333692 Scan the binary representation of n from left to right; at each 1, reverse the bits to the left and including this 1. The resulting binary representation is that of a(n).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 25, 20, 21, 26, 27, 24, 19, 22, 29, 28, 23, 30, 31, 32, 33, 34, 49, 36, 41, 50, 51, 40, 37, 42, 53, 52, 43, 54, 59, 48, 35, 38, 57, 44, 45, 58, 55, 56, 39, 46, 61, 60, 47, 62, 63, 64, 65, 66, 97

Offset: 0

Rémy Sigrist, Apr 04 2020

Leading zeros are ignored.
This sequence is a permutation of the nonnegative integers (as it is injective and preserves the binary length); see A333693 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 1 0 1)0 1 0
   (1 0 1 0 1 1)0
- the resulting binary representation is "1010110"
- and a(90) = 86.

See A333776 for a similar sequence.

Cf. A000120, A030101, A333693 (inverse), A333762 (fixed points).

a(n, base=2)={ my (b=digits(n, base), p=[]); for (k=1, #b, p=concat(p, b[k]); if (b[k], p=Vecrev(p))); fromdigits(p, base) }

a(2*n) = 2*a(n).
a(2*n+1) = A030101(2*a(n)+1).
A000120(a(n)) = A000120(n).

A333762 Fixed points of A333692.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 28, 30, 31, 32, 33, 34, 36, 40, 42, 48, 55, 56, 60, 62, 63, 64, 65, 66, 68, 72, 73, 80, 84, 85, 96, 103, 110, 112, 120, 124, 126, 127, 128, 129, 130, 132, 136, 144, 146, 160, 168, 170, 181

Offset: 1

Rémy Sigrist, Apr 04 2020

For any n >= 0, n belongs to this sequence iff 2*n also belongs to this sequence.

This sequence contains A000079, A000225, A018900, A023758, A295235.

Cf. A000079, A000225, A018900, A023758, A295235, A333692, A333693.

is(n, base=2)={ my (b=digits(n, base), p=[]); for (k=1, #b, p=concat(p, b[k]); if (b[k], p=Vecrev(p))); n==fromdigits(p, base) }

cp's OEIS Frontend

A333692 Scan the binary representation of n from left to right; at each 1, reverse the bits to the left and including this 1. The resulting binary representation is that of a(n).

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