I enjoy recreational mathematics, numbers, and logic puzzles. On this page you can find some of the mathematical curiosities I’ve found interesting.
Metaprimes
In 2003 I was playing around with mapping numbers encoded in normal, “additive” binary to numbers encoded in an alternate, “multiplicative” binary. You can read about it here.
Interesting integer sequences
The Online Encyclopedia of Integer Sequences is filled with other interesting (some would say beautiful) sequences; here are some of my favorites. I’ve also submitted a few sequences to the OEIS myself; there’s a list of them here.
Prime signatures
Prime signatures are a favorite mathematical topic of mine. Below are some posts related to them.
213
Since August 1 is the 213th day of the year (2023 not being a leap year), this seems like a good time to share an interesting fact about the number 213. It involves semiprimes, so let’s review what a semiprime is: a product of two distinct prime numbers. 213 is a semiprime because it is…
Prime signatures
Prime signatures are a way of describing the prime factorization of a number. The prime signature lists the exponents of the prime factors, typically sorted from smallest exponent to greatest (or vice versa). For example, the number 28’s prime factorization is 22 × 7, which can be abstracted to p2q (where p and q are distinct primes) or p12p2 (where p1 and p2 are distinct primes.) The prime…
Iterative mapping of prime signatures
Perhaps the most elegant way of arranging all of the positive integers in a two-dimensional grid is the pairing function. The grid is populated by placing a 1 in its starting corner, then filling the grid with diagonal stripes: 2 and 3 in the first stripe, 4 through 6 in the second, and so on. Below…
Prime signature table
A very large table of the first few numbers for each of the first 160 prime signatures.
List of prime signatures
Below is a list of the first 400 prime signatures. The OEIS column shows the Online Encyclopedia of Integer Sequences entry (if any) for the integers belonging to the specified prime signature. For example, row #8’s prime signature is {1,1,1}, so it links to OEIS sequence A007304, which lists integers with the prime signature of {1,1,1}. The smaller entries with…
Binary-encoded prime signatures
Prime signatures are a way of describing the prime factorization of a number. For example, the number 28’s prime factorization is 22 × 7, which can be abstracted to p2q (where p and q are distinct primes) or p12p2 (where p1 and p2 are distinct primes.) Encoding in binary Another way of expressing the prime factorization would be in to encode it as binary, where each binary digit represents…
More math posts:
Non-square-free numbers
A square-free number is one that does not contain a square as a factor. One interesting sequence involves non-square-free numbers: that is, numbers that are divisible by a square. The first number in this sequence is 4, which is the smallest integer containing a non-trivial (i.e. not 0 or 1) square. The second number is 8, which is the first…
Favorite mathematical sequences
Non-square-free numbers A square-free number is one that does not contain a square as a factor. One interesting sequence involves non-square-free numbers: that is, numbers that are divisible by a square. As of 2023 the highest number of consecutive non-square-free numbers is 18 according to the OEIS. Untouchable Aliquot Numbers Take any integer and add its proper divisors…
Fibonacci numbers and Pisano periods
The Fibonacci numbers (sequence A000045) form one of the most important and well-known sequences in mathematics. The sequence is very easy to construct: start with 0 and 1 as the first two numbers in the sequence, and after that, every new term in the sequence is the sum of the previous two: 0, 1, 1, 2, 3, 5,…
Online Encyclopedia of Integer Sequences entries
Relating to Metaprimes A079708: Metaprime binary to standard binary conversion series (first term = 0). A133487: Metaprime binary to standard binary conversion series (first term = 256). Fibonacci numbers and Pisano periods A179390: Modulus for Fibonacci-type sequence described by A015134. A179391: First term in Fibonacci-type sequence described by A015134. A179392: Second term in Fibonacci-type sequence described by…
Collatz tree / 3x+1 conjecture
The Collatz conjecture (also known as the “3x + 1” or “3n + 1” problem) involves a simple repeated formula, but yields some interesting sequences. In short, begin with a positive integer; if it’s even, divide it by two; if it’s odd, multiply it by three and add one. Repeat this sequence until the result is 1.…
Why does 0.999… equal 1?
One mathematical question that sometimes causes some confusion is this: “Are 0.999… and 1 the same thing?” The answer is yes, assuming by “0.999…” we mean a zero, then a decimal point, then nines that repeat forever. There are pages and pages of lengthy mathematical proofs available that show why this is so, but my favorite way of showing why 0.999……
Metaprimes
One day a few years ago, while waiting for my dinner, I started doodling with prime numbers, and I came across a question I found interesting. Here’s how it came about. When translating a number from binary to decimal notation by hand, you add together a series of products. For example: 1110 (binary) = (16…
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