A161001 Those positive integers n that when read in binary contains runs of 0's and 1's being of distinct lengths, the list of lengths forming a permutation of some number of consecutive positive integers.

1, 3, 4, 6, 7, 15, 24, 28, 31, 35, 39, 49, 55, 57, 59, 63, 112, 120, 127, 255, 391, 399, 451, 463, 480, 483, 487, 496, 511, 536, 540, 560, 572, 624, 632, 776, 782, 784, 798, 880, 888, 900, 902, 912, 926, 944, 956, 964, 966, 968, 974, 984, 988, 1023, 1984, 2016

Offset

1,2

Comments

Think of binary n as a string S of 0's and 1's. By a "run" of 0's or 1's, it is meant either a substring all of contiguous 0's, each run bounded by 1's or the edge of S; or a substring all of contiguous 1's, each run bounded by 0's or the edge of S.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

451 in binary is 111000011. This contains a run of three 1's, followed by a run of four 0's, followed by a run of two 1's. Since (3,4,2) is a permutation of some number of consecutive positive integers (2,3,4), then 451 is in the sequence.

Prog

(Python)
from itertools import groupby
def ok(n):
  runlengths = [len(list(g)) for k, g in groupby(bin(n)[2:])]
  minrl = min(runlengths)
  return sorted(runlengths) == list(range(minrl, minrl+len(runlengths)))
print([n for n in range(1, 2021) if ok(n)]) # Michael S. Branicky, Jan 04 2021
(Python) # alternate that directly generates terms
from itertools import permutations
def runlengths(k, r): # all terms with runlengths a permutation of k, ..., r
  c = ['1', '0']
  return sorted([int("".join([c[j%2]*p[j] for j in range(r-k+1)]), 2)
    for p in permutations(range(k, r+1))])
def aupto(nn):
  digits, k, r, out = 1, 1, 1, []
  while len(out) < nn:
    for r in range(1, digits + 1):
      for k in range(1, r + 1):
        if sum(range(k, r+1)) == digits:
          out += runlengths(k, r)
    digits += 1
  return sorted(set(out))[:nn]
print(aupto(56)) # Michael S. Branicky, Jan 04 2021

Crossrefs

Cf. A161000.

Sequence in context: A058639 A253888 A289811 * A168170 A290115 A225794

Adjacent sequences: A160998 A160999 A161000 * A161002 A161003 A161004

Keyword

base,nonn

Author

Leroy Quet, Jun 01 2009

Extensions

Extended by Ray Chandler, Jun 13 2009

Status

approved