A348111 Numbers k whose binary expansion starts with the concatenation of the binary expansions of the run lengths in binary expansion of k.

0, 1, 7, 14, 28, 112, 127, 254, 509, 1016, 1018, 1792, 2033, 2037, 2039, 4066, 4072, 4075, 4078, 8132, 8135, 8150, 8156, 16256, 16300, 16313, 32513, 32528, 32576, 32601, 32607, 32626, 32639, 32767, 65027, 65087, 65153, 65202, 65248, 65253, 65255, 65534, 130307

Offset: 1

Author

Rémy Sigrist, Oct 01 2021

Keywords

• nonn
• base

Comments

We consider here that 0 has an empty binary expansion, and include it in the sequence.

This sequence is infinite as it contains A077585.

Examples

			Regarding 32607:
- the binary expansion of 32607 is "111111101011111",
- the corresponding run lengths are: 7, 1, 1, 1, 5,
- in binary: "111", "1", "1", "1", "101",
- after concatenation: "111111101",
- as "111111101011111" starts with "111111101", 32607 belongs to this sequence.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A348111

Crossrefs

Cf. A077585, A101211, A175930, A319678.

Programs

PARI

See Links section.

Python

from itertools import groupby
def ok(n):
    if n == 0: return True
    b = bin(n)[2:]
    c = "".join(bin(len(list(g)))[2:] for k, g in groupby(b))
    return b.startswith(c)
print(list(filter(ok, range(2**17)))) # Michael S. Branicky, Oct 02 2021