Counting by prime factors and "base prime" notation

Question

Question is whether clear patterns exist in the $p_n$ "digit" of a factorization notation where each digit stands for the power of the prime in that digit's position. For example, one would count from 2 to 17 as follows:

1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4, 1000000

What pattern, if any, would we expect to see from the right-most digit in this notation (the power of 2 in the factorization). Or in the 2nd right-most digit (the power of 3 in the factorization).

If this notion pre-exists, would appreciate a pointer to it, as I am not sure how to describe it in an unknown unknown phraseology.

Answer 1

The digits in the $2$s place will look like the heights of marks on an imperial ruler, where the heights correspond to the power of $2$ in the denominator of a fraction. If we start counting at $1$ they start $$0,1,0,2,0,1,0,3,0,1,0,2,0,1,0$$ You can recursively generate the sequence. This is the order $3$ version as the central number is $3$. To make the order $n+1$ version take the $n$ version, append $n+1$ and append the $n$ version again. This is OEIS A007814 The construction of the sequence of digits in the $3$s place is similar. Start with $0,0$. Take three copies of the order $n$ version and put $n+1$ in the spaces between them. This is OEIS A007949. It starts $$0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0$$You should be able to extend this to each prime.

Answer 2

There is a problem in your proposed notation in that powers greater than 9 will eventually appear in the prime decomposition. Therefore certain numbers will have the same notation, such as 4096 and 12: $4096 = 2^{12}$ and $12 = 3^1 \times 2^2$. Interestingly, the number 12 corresponds to "12" in your notation.

Thus your notation will eventually be jumbled in that the nth position may not correspond to the power of the nth prime in the prime decomposition of the corresponding number.

Answer 3

I believe Ross Millikan is correct. But you did ask about notation (I think -- your last sentence says "notion" but the subject line said "notation"), as well. So I wanted to answer that part -- the "unknown phraseology" you mention :)

There's a handful of systems people have used for this number system; https://www.reddit.com/r/baseprime/comments/id341b/other_posts_about_prime_notation_systems_base/ has some links to some (https://www.reddit.com/r/puremathematics/comments/f7ayn0/base_prime_written_in_base_prime/ and https://www.reddit.com/r/askscience/comments/58i1ww/would_a_baseprime_number_system_be_useful_in_any/ are maybe the two most interesting links there) but I've always favored using list notation for this, to relieve theambiguity pointed out by agarttha: 1 -> {} 2 -> {1} 3 -> {0,1} 4 -> {2} 5 -> {0,0,1} etc.

I really want to submit a comment, too, suggesting alternate ways of looking for patterns, in case that would be helpful for whatever topic you are investigating, but you have to have a reputation of 50 to submit a comment... so I'm just going to leave this here as an addendum on my answer submission:

How about putting the natural numbers into a (nearly) binary tree? Instead of constructing the "base prime" numbers from the naturals, go the other way around. If you are looking for a relationship between a sequence of numbers and patterns in complex representations, sometimes it is helpful to approach from the opposite direction: create patterns in those complex representations which are bijective with your original sequence, and then work backwards to see how that maps to your original sequence.

For example, in "base prime," start with 1 and define two operations: (1) adding a "0" to the end (so shifting every prime exponent up to the next-higher prime factor), (2) increasing the first digit by 1 (raising the exponent in the 2's place). If you build a tree this way, where each node has a right branch (the 0 concatenation) and a left branch (the increment of the 2's place), you'll (eventually) get all whole numbers greater than or equal to 2. I have a diagram where I did a few rows of this, to help explain it visually; note that I used the aforementioned list notation (and threw in a 0 at the top, which is the "base prime" representation of 1), but you can view it at http://copland.udel.edu/~os/baseprime-bintree-ljpost-binarytree.png to see it. Once you have this binary tree constructed, just walk horizontally across each row from top to bottom and you'll have an ordering which does some odd, pattern-like things, but not completely. It starts out (if you walk from right to left) 2; 3, 4; 5, 6, 9, 8: 7, 10, 15, 12, 25, 18, 27, 16; 11, 14, 21, 20, 35, 30, 45, 24, 49, 50, 75, 36, 125, 54, 81, 32; ... (and you can calculate the mean of each row, for example: 2; 3.5; 7; 16.25; 43.875; ... which seem to follow a pattern of semi-steadily diverging ratios -- something which could warrant further study)