File:OR and XOR of binary exponents.svg

Each natural number (non-negative integer) can be assigned its set of binary exponents.

Each of these sets can be assigned the bitwise OR and XOR of its elements.
The exponent 0 has no effect in this operation, so the resulting sequences contain pairs of duplicate entries. This duplication has been removed.

The XOR sequence partitions the first ${\displaystyle 2^{2^{n}}}$ natural numbers into ${\displaystyle 2^{n}}$ blocks of equal size. The first 256 entries are partitioned into 8 blocks of size 32.

The OR sequence contains all natural numbers, but each one only a finite number of times.
Most entries among the first ${\displaystyle 2^{2^{n}}}$ are ${\displaystyle 2^{n}-1}$. Among the first 256 entries the 7 appears 218 times.
An entry with binary weight ${\displaystyle k}$ appears  A000371(${\displaystyle k}$) times. That is the number of non-degenerate ${\displaystyle k}$-ary Boolean functions.

The duplication has been removed, so the entries corresponding to the first 16 natural numbers are described by the first 8 entries shown here:

OR:  A327041 = 0, 1, 2, 3,   3, 3, 3, 3,   4, 5, 6, 7,   7, 7, 7, 7...

XOR:  A261283 = 0, 1, 2, 3,   3, 2, 1, 0,   4, 5, 6, 7,   7, 6, 5, 4...   ( A253315 is the same with duplicated entries.)

The big matrix contains scatterplots of the three sequences.
Black dots are the OR sequence. Red and green rings are XOR and the difference.