A359482 -id:A359482 - cp's OEIS frontend

A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).

1, 10, 98, 767, 111, 122, 2, 11, 100, 889, 110, 4490, 400, 560, 1096, 124, 20, 129, 70, 502, 93, 171, 212, 361, 512, 26, 21, 36, 54, 14, 1011, 139, 99, 59, 550, 684, 345, 102, 1021, 1999, 2871, 137, 892, 89, 126, 875, 510, 994, 586, 2012, 662, 1836, 201, 405, 388, 2007, 2798, 1641, 50, 340

Offset: 1

Scott R. Shannon and Eric Angelini, Jul 07 2023

In the first 10000 terms the smallest number that has not yet appeared is 696; it is therefore likely all numbers eventually appear although this is unknown.

			a(2) = 10 as a(1) + 10 = 1 + 10 = 11 which is a substring of "1" + "10" = "110".
a(3) = 98 as a(1) + a(2) + 98 = 1 + 10 + 98 = 109 which is a substring of "1" + "10" + "98" = "11098".
a(4) = 767 as a(1) + a(2) + a(3) + 767 = 1 + 10 + 98 + 767 = 876 which is a substring of "1" + "10" + "98" + "767" = "11098767".

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Eric Angelini, Échecs et Maths, Personal blog, July 2023.

Cf. A359482, A300000, A339144, A357082.

from itertools import islice
def agen(): # generator of terms
    s, mink, aset, concat = 1, 2, {1}, "1"
    yield from [1]
    while True:
        an = mink
        while an in aset or not str(s+an) in concat+str(an): an += 1
        aset.add(an); s += an; concat += str(an); yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Feb 08 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

A363186 Lexicographically earliest sequence of distinct positive integers such that the sum of all terms a(1)..a(n) is a substring of the concatenation of all terms a(1)..a(n).

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