File:OR and XOR of binary exponents.svg

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Description

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 natural numbers into 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 are . Among the first 256 entries the 7 appears 218 times.
An entry with binary weight appears Sloane'sA000371() times. That is the number of non-degenerate -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: Sloane'sA327041 = 0, 1, 2, 3,   3, 3, 3, 3,   4, 5, 6, 7,   7, 7, 7, 7...

XOR: Sloane'sA261283 = 0, 1, 2, 3,   3, 2, 1, 0,   4, 5, 6, 7,   7, 6, 5, 4...   (Sloane'sA253315 is the same with duplicated entries.)

The entries of the XOR sequence are bitwise smaller than those in the OR sequence. Their difference is shown for the sake of completeness.
Text versions of the shown sequences can be found here.


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

This image illustrates only the OR sequence.
The colors shows if the dot is on a light field (pink) or a dark field (blue).
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Watchduck
You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".
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version without scatterplots

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Date/TimeThumbnailDimensionsUserComment
current22:36, 21 May 2024Thumbnail for version as of 22:36, 21 May 20242,780 × 680 (34 KB)Watchduck (talk | contribs)circle markers
14:50, 21 May 2024Thumbnail for version as of 14:50, 21 May 20242,750 × 600 (22 KB)Watchduck (talk | contribs)detail
11:37, 21 May 2024Thumbnail for version as of 11:37, 21 May 20242,750 × 600 (22 KB)Watchduck (talk | contribs)Uploaded own work with UploadWizard

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