A354774 For terms of A354169 that are the sum of two distinct powers of 2, the exponent of the larger power of 2.

1, 3, 4, 5, 6, 7, 9, 10, 4, 11, 13, 14, 6, 15, 17, 18, 19, 20, 21, 22, 5, 23, 25, 26, 27, 28, 29, 30, 7, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 23, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 31, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Author

N. J. A. Sloane, Jun 26 2022

Keywords

• nonn

Comments

Taking first differences, then applying the RUNS transform gives [1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 13, 1, 1, 1, 13, 1, 1, 1, 29, 1, 1, 1, 29, 1, 1, 1, 61, 1, 1, 1, 61, 1, 1, 1, 125, 1, 1, 1, 125, 1, 1, 1, 253, 1, 1, 1, 253, 1, 1, 1, 509, ...].

If the initial 4 is changed to a 1, this has an obvious regular structure, which could then be analyzed to give a conjectured generating function, just as was done for A354767. See link below.

A more precise conjecture is given in the Formula section.

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
• N. J. A. Sloane, A conjectured generating function for A354169.

Crossrefs

Cf. A354169, A354680, A354767, A354798, A354773, A354775.

Programs

Python

from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354774_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)[3:]).count('1') == 1:
                    yield len(s)
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_,aqueue)
                f = not f
                break
A354774_list = list(islice(A354774_gen(),30)) # Chai Wah Wu, Jun 27 2022

Formula

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

The following is a conjectured explicit formula for a(n). Basically a(n) = n+2, except that there are four types of n which have a different formula, and there are 6 exceptional values for small n.

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

(I) If n = 3*2^(k-1)-3, k >= 2 then a(n) = (n+1)/2, except a(3) = a(9) = 4 and a(21) = 5.

(II) If n = 2^(k+1)-3, k >= 1 then a(n) = (n+1)/2, except a(5) = a(13) = 6 and a(29) = 7.

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

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

(V) Otherwise a(n) = n+2. (End)

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