A357082 -id:A357082 - cp's OEIS frontend

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.

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.

A357482 a(0) = 0; for n > 0, a(n) is the smallest positive number not occurring earlier such that the binary string of the number of 1's in the binary value of a(n) + the number of 1's in the binary values of all previous terms does not appear in the binary string concatenation of a(0)..a(n-1).

Original entry on oeis.org

0, 1, 2, 3, 7, 4, 5, 63, 8, 6, 9, 16, 127, 11, 10, 12, 13, 14, 19, 511, 1023, 15, 21, 17, 31, 18, 20, 22, 24, 25, 33, 23, 27, 26, 28, 35, 37, 38, 41, 1535, 29, 30, 32, 34, 47, 36, 40, 55, 39, 43, 42, 45, 255, 46, 51, 383, 48, 44, 4095, 64, 447, 65, 95, 53, 191, 767, 1791, 59, 49, 54, 57, 50, 52

Offset: 0

Scott R. Shannon, Sep 30 2022

The sequence contains large jumps in value due to some terms having to be 1 less than a power of 2 to contain sufficient 1's in their binary value to meet the term selection criteria. For example a(386) = 512, a(387) = 68719476735. See the examples below.

			a(7) = 63 as 63 = 111111_2 which contains six 1's, the concatenation of the binary values of a(0)..a(6) is "011011111100101" which contains ten 1's, and 6 + 10 = 16 = 10000_2 which does not appear in the concatenated binary string of previous terms. All smaller unused numbers less than 63 have one to five 1's in their binary values leading to sums of 11, 12, 13, 14 or 15, but the binary values of these five sums all appear in the concatenated binary string of previous terms.

Cf. A357082, A357449, A355611, A007088, A030302, A118248.

cp's OEIS Frontend

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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