A354798 Indices of terms in A354169 that are not powers of 2.

0, 5, 9, 13, 15, 19, 21, 25, 29, 31, 33, 37, 41, 43, 45, 49, 53, 55, 59, 61, 65, 67, 69, 73, 77, 79, 83, 85, 89, 91, 93, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 141, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 175, 179

Offset: 1

Author

Rémy Sigrist and N. J. A. Sloane, Jun 06 2022

Keywords

• nonn
• base

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] [math.CO], Sep 11 2022.
• Rémy Sigrist, PARI program

Crossrefs

Cf. A057716, A136252, A354169, A354680 (corresponding terms), A354788, A354798.

Programs

PARI

See Links section.

Python

from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354798_gen(): # generator of terms
    aset, aqueue, b, f, i = {0,1,2}, deque([2]), 2, False, 2
    yield 0
    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:
                i += 1
                if m.bit_count() > 1:
                    yield i
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_,aqueue)
                f = not f
                break
A354798_list = list(islice(A354798_gen(),30)) # Chai Wah Wu, Jun 06 2022

Formula

Conjecture from N. J. A. Sloane, Jul 15 2022: (Start)

The following is a conjectured explicit formula for a(n).

Define the "fence posts" by F(0) = 1, F(2i+1) = 2^(i+4) - 3 for i >= 0, F(2i) = 3*2^(i+2) - 3 for i >= 1.

The F(i) sequence begins 1, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, ... (cf. A136252 or A354788)

The value of a(n) at n = F(i) is V(i) = 0 if i = 0, V(i) = 3*F(i)+2 if i >= 1.

The V(i) sequence begins 0, 41, 65, 89, 137, 185, 281, 377, 569, 761, ... (cf. A354789).

The first 12 terms of the sequence are irregular, so we simply define a(n) for F(0) = 1 <= n <= 12 to be the n-th term of

[0, 5, 9, 13, 15, 19, 21, 25, 29, 31, 33, 37]

Assume now that n >= F(1) = 13, and define i and j by F(i) <= n < F(i+1), n = F(i) + j.

Then we conjecture that a(n) = V(i) + f(j) where f(0) .. f(3) are 0,2,4,8, and for j >= 4, f(j) = 3*j if j is even, f(j) = 3*j-1 if j is odd.

The f(i), i >= 0, sequence is independent of n (to find a(n) we use only an initial segment of f(n)), and begins:

0, 2, 4, 8, 12, 14, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50, 54, 56, ...

The conjecture has been checked for the first 5000 terms.

(End)

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