A363186 -id:A363186 - cp's OEIS frontend

A369899 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that is a substring of the sum of all previous terms.

1, 2, 3, 6, 12, 4, 8, 36, 7, 9, 88, 17, 19, 21, 23, 5, 26, 28, 15, 30, 60, 20, 40, 48, 52, 58, 38, 67, 43, 78, 64, 92, 10, 103, 11, 14, 115, 27, 13, 31, 34, 37, 41, 45, 50, 51, 16, 18, 63, 69, 68, 83, 91, 201, 22, 33, 66, 32, 236, 260, 86, 29, 75, 305, 35, 39, 42, 47, 351, 386, 25, 80, 360, 72

Offset: 1

Scott R. Shannon, Feb 05 2024

The fixed points begin 1, 2, 3, 94, 1420, 1423, 1425, 1426, 1427, 8592, although it is likely there are infinitely more. The sequence is conjectured to be a permutation of the positive numbers.

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

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

Cf. A370046 (base 2), A363186, A333410.

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, str(s)
        while an in aset or not str(an) in ss: an += 1
        aset.add(an); s += an; yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Feb 08 2024

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.

Original entry on oeis.org

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

Scott R. Shannon, Feb 08 2024

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.

			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".

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

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

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

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

A364201 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) in binary is a substring of the concatenation of all terms a(1)..a(n) in binary.

Original entry on oeis.org

1, 2, 3, 5, 11, 7, 16, 9, 6, 18, 4, 13, 10, 15, 12, 23, 20, 8, 27, 19, 36, 26, 22, 17, 21, 31, 25, 14, 29, 28, 30, 57, 24, 32, 39, 43, 40, 34, 38, 46, 33, 35, 42, 37, 55, 44, 58, 48, 56, 52, 41, 45, 64, 63, 54, 61, 60, 49, 50, 51, 65, 47, 67, 88, 132, 73, 76, 68, 109, 59, 82, 87, 62, 98, 69, 70

Offset: 1

Scott R. Shannon, Jul 13 2023

In the first 10000 terms the smallest number that has not yet appeared is 7026; it is conjectured all numbers eventually appear.

The fixed points begin 1, 2, 3, 29, 48, 68, 96, 182, 471, 839, ... . It is likely there are infinitely more.

			a(2) = 2 as a(1) + 2 = 1 + 2 = 3 = 11_2, which is a substring of "a(1)"_2 + "2"_2 = "1" + "10" = "110".
a(4) = 5 as a(1) + a(2) + a(3) + 5 = 1 + 2 + 3 + 5 = 11 = 1011_2, which is a substring "a(1)"_2 + "a(2)"_2 + "a(3)"_2 + "5"_2 = "1" + "10" + "11" + "101" = "11011101".
a(5) = 11 as a(1) + a(2) + a(3) + a(4) + 11 = 1 + 2 + 3 + 5 + 11 = 22 = 10110_2, which is a substring "a(1)"_2 + "a(2)"_2 + "a(3)"_2 + "a(4)"_2 + "11"_2 = "1" + "10" + "11" + "101" + "1011" = "110111011011".

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

Cf. A363186 (base 10), A359482, A007088, A300000, A339144, A357082.

cp's OEIS Frontend

A369899 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that is a substring of the sum of all previous terms.

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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.

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A364201 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) in binary is a substring of the concatenation of all terms a(1)..a(n) in binary.

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