A166751 Even positive integers in which, when written in binary, each run of 0's is of exactly the same length as the run of 1's immediately before it.

2, 10, 12, 42, 44, 50, 56, 170, 172, 178, 184, 202, 204, 226, 240, 682, 684, 690, 696, 714, 716, 738, 752, 810, 812, 818, 824, 906, 908, 962, 992, 2730, 2732, 2738, 2744, 2762, 2764, 2786, 2800, 2858, 2860, 2866, 2872, 2954, 2956, 3010, 3040, 3242, 3244, 3250, 3256

Offset: 1

Author

Leroy Quet, Oct 21 2009

Keywords

• base
• nonn
• changed

Comments

Each term, when written in binary, contains an even number of digits, obviously.

The runs of 0's in the n-th term form the composition in the n-th row of A228369. - John Tyler Rascoe, Sep 05 2024

Examples

			The first 7 terms written in binary: 10, 1010, 1100, 101010, 101100, 110010, 111000.
From _Paolo Xausa_, Aug 28 2025: (Start)
Terms can be arranged in an irregular triangle, where row n >= 1 has length 2^(n-1), row sum A386705(n), and lists all the terms with bit length 2*n:
    2;
   10,  12;
   42,  44,  50,  56;
  170, 172, 178, 184, 202, 204, 226, 240;
  682, 684, 690, 696, 714, 716, 738, 752, 810, 812, 818, 824, 906, 908, 962, 992;
  ... (End)
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..8192

Crossrefs

Cf. A066099, A164707, A175413, A228369, A386705.

Programs

Mathematica

A166751row[n_] := With[{b = Array[IntegerDigits[4^# - 2^#, 2] &, n]}, Sort[Flatten[Map[FromDigits[Flatten[#], 2] &, Map[b[[#]] &, Map[Permutations, IntegerPartitions[n]], {2}], {2}]]]]; (* Generates terms with bit length = 2*n *)
Array[A166751row, 6] (* Paolo Xausa, Aug 28 2025 *)

Python

from itertools import groupby
def ok(n):
    L = [len(list(g)) for k, g in groupby(bin(n)[2:])]
    return (m:=len(L))&1 == 0 and all(L[2*j] == L[2*j+1] for j in range(m>>1))
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Aug 25 2025

Extensions

Extended by Ray Chandler, Mar 11 2010