A308811 Numbers k such that the binary plot of the list of divisors of k has reflection symmetry.

1, 2, 3, 4, 8, 10, 15, 16, 32, 64, 128, 136, 170, 255, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 32896, 34952, 43690, 65535, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728

Offset: 1

Author

Rémy Sigrist, Jul 08 2019

Keywords

• nonn
• base

Comments

The sequence is infinite as it contains every power of 2 (A000079).

The product of the first five Fermat primes (A019434), 4294967295 = 3 * 5 * 17 * 257 * 65537, is also a member of this sequence.

Every term belongs to A135772.

The first 48 terms are all of the form Sum_{i=1..t} 2^(k*t-1) for some k > 0 and t > 0 (see binary plot in Links section).

Examples

			Regarding 170:
- the divisors of 170 are: 1, 2, 5, 10, 17, 34, 85, 170,
- in binary: "1", "10", "101", "1010", "10001", "100010", "1010101", "10101010",
- the corresponding binary plot is:
  .             1         .             #
    .         1 0           .         #
      .     1 0 1             .     #   #
        . 1 0 1 0               . #   #
        1 0 0 0 1               # .     #
      1 0 0 0 1 0             #     . #
    1 0 1 0 1 0 1           #   #   # . #
  1 0 1 0 1 0 1 0         #   #   #   # .
                  .                       .
                    .                       .
- this binary plot has reflection symmetry,
- hence 170 belongs to this sequence.
		

Links

• Rémy Sigrist, Binary plot of the first 48 terms

Crossrefs

Cf. A000079, A019434, A135772, A295368.

Programs

PARI

is(n) = { my (d=Vecrev(divisors(n))); if (#binary(d[1])==#d, for (b=0, #d-1, my (t=0); for (i=1, #d, if (bittest(d[i], b), t+=2^(i-1))); if (t!=d[b+1], return (0))); return (1), return (0)) }

Formula

A295368(a(n)) = a(n).