Category: prime signatures

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 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…

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…

A very large table of the first few numbers for each of the first 160 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…

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…