A370046 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number whose binary value is a substring of the binary value of the sum of all previous terms.

1, 2, 3, 6, 4, 8, 12, 9, 5, 18, 17, 10, 7, 19, 14, 16, 11, 20, 13, 24, 22, 15, 32, 36, 34, 25, 23, 37, 27, 21, 26, 64, 69, 40, 43, 29, 30, 35, 39, 44, 28, 42, 53, 129, 72, 38, 31, 81, 45, 50, 46, 47, 49, 74, 41, 54, 55, 51, 52, 57, 58, 128, 68, 70, 140, 77, 60, 139, 85, 33, 75, 61, 59, 62, 48

Offset: 1

Author

Scott R. Shannon, Feb 08 2024

Keywords

• nonn
• base

Comments

The fixed points begin 1, 2, 3, 16, 39, 42, 50, 79, 120, 361, although it is likely there are infinitely more. The sequence is conjectured to be a permutation of the positive numbers.

Examples

			a(7) = 12 as the sum of all previous terms is 1 + 2 + 3 + 6 + 4 + 8 = 24 = 11000_2 and 12 = 1100_2 is the smallest unused number that is a substring of "11000".
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 50000 terms.

Crossrefs

Cf. A317788, A369899 (base 10), A363186, A333410.

Programs

Python

from itertools import islice
def agen(): # generator of terms
    s, mink, aset = 3, 3, {1, 2}
    yield from [1, 2]
    while True:
        an, ss = mink, bin(s)[2:]
        while an in aset or not bin(an)[2:] in ss: an += 1
        aset.add(an); s += an; yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 75))) # Michael S. Branicky, Feb 08 2024

Formula

a(n) = A317788(n) for any n >= 3. - Rémy Sigrist, Feb 09 2024