A366139 -id:A366139 - cp's OEIS frontend

A366140 Fixed points of the binary rotations A336953 and A366139: numbers k >= 0 such that A336953(k) = A366139(k) = k.

0, 1, 2, 3, 6, 7, 8, 10, 12, 15, 20, 25, 30, 31, 36, 42, 45, 48, 54, 60, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 127, 128, 136, 144, 152, 160, 168, 170, 176, 184, 192, 200, 204, 208, 216, 224, 232, 240, 248, 255, 261, 270, 279, 288, 297, 306, 315, 324, 333

Offset: 1

Paolo Xausa, Sep 30 2023

If a number is a fixed point of A336953, then it's also a fixed point of A366139, and vice versa.

k is a term iff A302291(k)|k.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A302291, A336953, A366139.

A366140Q[n_]:=FromDigits[RotateLeft[IntegerDigits[n,2],n],2]==n;
Select[Range[0,500],A366140Q]

A336953 In binary representation, rotate the digits of n right n places.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 7, 8, 12, 10, 7, 12, 14, 11, 15, 8, 12, 10, 7, 20, 26, 21, 30, 17, 25, 13, 30, 19, 27, 30, 31, 8, 12, 10, 7, 36, 50, 41, 60, 34, 19, 42, 53, 11, 45, 58, 31, 48, 56, 44, 30, 19, 43, 54, 59, 14, 15, 43, 55, 60, 62, 47, 63, 32, 48, 40, 28

Offset: 0

Gage Schacher, Aug 08 2020

On the graph, there are a series of larger and larger parallelograms joined together by a straight line on y=x where n is unchanged, mostly in the case where n is a multiple of the bit length of n. In addition to the main line that cuts through the graph, each parallelogram has the same few sloped lines in its borders.

			a(3) = a('11') = '11' = 3;
a(4) = a('100') = '010' = '10' = 2;
a(5) = a('101') = '011' = '11' = 3;

Rémy Sigrist, Table of n, a(n) for n = 0..16384

Cf. A007088, A038572 (rotated one binary place to the right).

Cf. A366139 (rotate left), A366140 (fixed points).

Array[FromDigits[RotateRight[IntegerDigits[#, 2], #], 2] &, 68, 0] (* Michael De Vlieger, Oct 05 2020 *)

a(n) = my(d=binary(n)); for (k=1, n, d = concat(d[#d], d[1..#d-1])); fromdigits(d, 2); \\ Michel Marcus, Aug 09 2020

def A336953(n):
    if n == 0: return 0
    l, m = -(n%n.bit_length()), bin(n)[2:]
    return int(m[l:]+m[:l],2) # Chai Wah Wu, Jan 22 2023

cp's OEIS Frontend

A366140 Fixed points of the binary rotations A336953 and A366139: numbers k >= 0 such that A336953(k) = A366139(k) = k.

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