A370046 - OEIS

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

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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