A354680 Terms of A354169 that are not powers of 2, in order of appearance.

0, 3, 12, 17, 34, 68, 136, 768, 1025, 18, 2080, 12288, 16388, 72, 32896, 196608, 262400, 524800, 1048577, 2098176, 4194306, 48, 8390656, 50331648, 67112960, 134225920, 268435460, 536887296, 1073741832, 192, 2147516416, 12884901888, 17179934720, 34359869440

Offset: 1

Author

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

Keywords

• nonn
• base

Comments

Apart from the initial 0, all terms have Hamming weight 2. See De Vlieger et al. (2022). - N. J. A. Sloane, Aug 29 2022

Examples

			The initial terms of A354169 are:
  0, 1, 2, 4, 8, 3, 16, 32, 64, 12, 128, 256.
The initial terms of this sequence are therefore: 0,             3,             12.
and the initial terms of A354798 are
  0,             5,              9.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..3320
• 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, PARI program
• Rémy Sigrist, PARI program (optimized version)

Crossrefs

Cf. A000120, A057716, A354169, A354798 (corresponding indices).

See also A354767.

Programs

PARI

See Links section.
(Python 3.10+)
from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354680_gen(): # generator of terms
    aset, aqueue, b, f = {0,1,2}, deque([2]), 2, False
    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:
                if m.bit_count() > 1:
                    yield m
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_,aqueue)
                f = not f
                break
A354680_list = list(islice(A354680_gen(),40)) # Chai Wah Wu, Jun 06 2022

Formula

A000120(A354169(a(n))) <> 1.