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

1,2

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.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the first 50000 terms.

Formula

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

Example

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".

Prog

(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

Crossrefs

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

Sequence in context: A349702 A289055 A109890 * A373326 A086537 A212486

Adjacent sequences: A370043 A370044 A370045 * A370047 A370048 A370049

Keyword

nonn,base

Author

Scott R. Shannon, Feb 08 2024

Status

approved