A333400 -id:A333400 - cp's OEIS frontend

A333398 Partial sums of A333400.

0, -1, -3, -2, -5, -9, -7, -4, 1, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 33, 23, 12, 25, 39, 27, 14, 29, 45, 31, 16, 34, 18, 35, 54, 37, 57, 38, 20, 41, 63, 43, 22, 46, 24, 47, 72, 49, 75, 51, 26, 53, 81, 55, 28, 58, 30, 59, 90, 61, 93, 62, 32, 65, 99, 67

Offset: 1

Alec Jones, Mar 18 2020

Conjecture: This is a permutation of the integers.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A333400.

A333401 a(n) is the distance between n and -n in A333400.

Original entry on oeis.org

2, 4, 3, 4, 3, 3, 3, 3, 3, 4, 3, 5, 3, 5, 3, 4, 2, 7, 3, 5, 3, 4, 2, 6, 4, 5, 3, 4, 2, 7, 3, 5, 4, 5, 4, 7, 2, 5, 4, 4, 2, 6, 4, 5, 6, 3, 4, 6, 2, 5, 3, 5, 3, 7, 3, 5, 5, 4, 4, 5, 4, 5, 4, 4, 2, 7, 3, 5, 3, 5, 4, 7, 3, 5, 4, 4, 2, 7, 3, 5, 6, 3, 4, 6, 2, 5, 3

Offset: 1

Alec Jones, Mar 18 2020

nonn

Records occur at n = 1, 2, 12, 18, 108, 216, 8856, ...

Interestingly, a(224) = a(225) = ... = a(441) = 3.

Peter Kagey, Table of n, a(n) for n = 1..10000

A367262 Lexicographically earliest sequence of distinct nonnegative integers such that the values a(0) XOR ... XOR a(k) (for some k >= 0) are all distinct (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 7, 8, 5, 14, 9, 16, 10, 11, 12, 15, 13, 25, 17, 18, 19, 21, 22, 20, 24, 29, 23, 32, 26, 27, 28, 30, 31, 49, 33, 35, 34, 37, 38, 41, 40, 36, 44, 64, 39, 42, 43, 45, 46, 47, 48, 50, 51, 52, 54, 53, 56, 55, 59, 57, 60, 58, 66, 65, 69, 67, 61, 96

Offset: 0

Rémy Sigrist, Nov 11 2023

This sequence is a variant of A333400; here we combine initial terms with the XOR operator, there with the addition.
This sequence is well defined; after some initial terms we can extend the sequence with a power of 2 greater that any prior term or even a smaller value.
This sequence is a permutation of the nonnegative integers (with inverse A367263):
- for any k >= 0, the least value >= 2^k is precisely 2^k,
- all powers of 2 appear in the sequence,
- after a power of 2, if the least value not yet in the sequence is less than this power of 2, then this value will be the next term.

			The first terms are:
  n   a(n)  a(0) XOR ... XOR a(n)
  --  ----  ---------------------
   0     0                      0
   1     1                      1
   2     2                      3
   3     4                      7
   4     3                      4
   5     6                      2
   6     7                      5
   7     8                     13
   8     5                      8
   9    14                      6
  10     9                     15
  11    16                     31
  12    10                     21

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, C++ program
Index entries for sequences that are permutations of the natural numbers

Cf. A333400, A346298, A367263 (inverse), A367264.

cp's OEIS Frontend

A333398 Partial sums of A333400.

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A333401 a(n) is the distance between n and -n in A333400.

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A367262 Lexicographically earliest sequence of distinct nonnegative integers such that the values a(0) XOR ... XOR a(k) (for some k >= 0) are all distinct (where XOR denotes the bitwise XOR operator).

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