A360320 Numbers k such that the total number of consecutive runs of zeros of length m in every binary expansion from 1 to k, is even, for all m != floor(log_2(k)).

1, 2, 3, 5, 11, 20, 21, 22, 42, 43, 82, 85, 162, 171, 322, 340, 341, 342, 642, 682, 683, 1282, 1362, 1365, 2562, 2722, 2731, 5122, 5442, 5460, 5461, 5462, 10242, 10882, 10922, 10923, 20482, 21762, 21842, 21845, 40962, 43522, 43682, 43691, 81922, 87042, 87362

Offset

1,2

Comments

The longest run of zeros in all binary expansions from 1 to k will belong to the largest power of 2 less than or equal to k, if it exists. The run will be of length m_max = floor(log_2(k)), and exactly one run of length m_max will exist for any k > 1.

Inclusion in this sequence is a prerequisite for forming a palindrome from the binary expansions of 1 to k, described in A291633.

Links

Table of n, a(n) for n=1..47.

Example

k=1 is a term since 1..k in binary is 1 alone which has no zero runs, therefore all numbers of runs are even.
k=2 is a term since 1..k in binary is 1.10 which is 1 run of 1 zero which is length m_max = floor(log_2(k)) = 1.
k=3:  1.10.11          # 1 run of 1, m_max = 1
k=5:  1.10.11.100.101  # 2 runs of 1, 1 run of 2, m_max = 2
k=11: 1.10.11.100.101.110.111.1000.1001.1010.1011
which is 6 of 1, 2 of 2, 1 of 3, m_max = 3
k=20: 1.10.11.100.101.110.111.1000.1001.1010.1011.1100.
      1101.1110.1111.10000.10001.10010.10011.10100
which is 10 of 1, 6 of 2, 2 of 3, 1 of 4, m_max = 4

Prog

(Python)
from collections import Counter
from itertools import count, groupby, islice
def rlz(n): return [len(list(g)) for k, g in groupby(bin(n)[2:]) if k=="0"]
def agen(): # generator of terms
    c = Counter()
    for n in count(1):
        c.update(r for r in rlz(n))
        if all(c[m]&1 == 0 for m in c if m != max(c)):
            yield n
print(list(islice(agen(), 50))) # Michael S. Branicky, Feb 04 2023

Crossrefs

Cf. A291633 (a subsequence).

Sequence in context: A358717 A337347 A087581 * A049888 A117221 A073879

Adjacent sequences: A360317 A360318 A360319 * A360321 A360322 A360323

Keyword

nonn

Author

Tyler Busby, Feb 03 2023

Status

approved