A354773 For terms of A354169 that are the sum of two distinct powers of 2, the exponent of the smaller power of 2.

0, 2, 0, 1, 2, 3, 8, 0, 1, 5, 12, 2, 3, 7, 16, 8, 9, 0, 10, 1, 4, 11, 24, 12, 13, 2, 14, 3, 6, 15, 32, 16, 17, 8, 18, 9, 19, 0, 20, 10, 21, 1, 22, 4, 5, 11, 48, 24, 25, 12, 26, 13, 27, 2, 28, 14, 29, 3, 30, 6, 7, 15, 64, 32, 33, 16, 34, 17, 35, 8, 36, 18, 37, 9, 38, 19, 39, 0, 40, 20, 41, 10, 42, 21, 43, 1, 44, 22, 45, 4, 46

Offset

1,2

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.

Rémy Sigrist, C++ program

Formula

Conjecture from N. J. A. Sloane, Jun 29 2022: (Start)

The following is a conjectured recurrence for a(n). Basically a(n) = a(n/2-1) if n is even, and a(n) = (n+1)/2 if n is odd, except that there are four types of n which have a different formula, and there are 19 exceptional values for small n. Note that a(n) does not depend on earlier values when n is odd.

Here is the formula, which agrees with the first 10000 terms.

There are exceptional values as far out as n=61, so we take care of them first.

Initial conditons:

If n is on the list

[1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 21, 22, 29, 30, 45, 61]

then a(n) is given by the n-th term of the following list:

[0, 2, 0, 1, 2, 3, 8, 0, 1, 5, 12, 2, 3, 7, 16, 8, 9, 0, 10,

1, 4, 11, 24, 12, 13, 2, 14, 3, 6, 15, 32, 16, 17, 8, 18, 9,

19, 0, 20, 10, 21, 1, 22, 4, 5, 11, 48, 24, 25, 12, 26, 13,

27, 2, 28, 14, 29, 3, 30, 6, 7].

Otherwise, if n is even, a(n) = a(n/2-1).

Otherwise n is odd and is not one of the exceptions.

(I) If n = 3*2^k-3, k >= 5, then a(n) = (n-1)/4.

(II) If n = 2^k-3, k >= 4 then a(n) = (n-1)/4.

(III) If n = 3*2^k-1, k >= 2 then a(n) = n+1.

(IV) If n = 2^k-1, k >= 3 then a(n) = n+1.

(V) Otherwise a(n) = (n+1)/2.

(End)

The conjecture is now known to be true. See De Vlieger et al. (2022). - N. J. A. Sloane, Aug 29 2022

Prog

(Python)
from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354773_gen(): # generator of terms
    aset, aqueue, b, f = {0, 1, 2}, deque([2]), 2, False
    while True:
        for k in count(1):
            m, j, j2, r, s = 0, 0, 1, b, k
            while r > 0:
                r, q = divmod(r, 2)
                if not q:
                    s, y = divmod(s, 2)
                    m += y*j2
                j += 1
                j2 *= 2
            if s > 0:
                m += s*2**b.bit_length()
            if m not in aset:
                if (s := bin(m)[:1:-1]).count('1') == 2:
                    yield s.index('1')
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_, aqueue)
                f = not f
                break
A354773_list = list(islice(A354773_gen(), 20)) # Chai Wah Wu, Jun 26 2022
(C++) See Links section.

Crossrefs

Cf. A354169, A354680, A354767, A354798, A354774, A354775.

Sequence in context: A115218 A023858 A011118 * A368724 A304784 A354665

Adjacent sequences: A354770 A354771 A354772 * A354774 A354775 A354776

Keyword

base,nonn

Author

N. J. A. Sloane, Jun 26 2022

Status

approved