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 together to produce a new number. Repeat the process with the new number and you’ve got an Aliquot sequence.

Some numbers will never appear in an aliquot sequence except at the beginning; these are the untouchable numbers. The first few are 2, 5, 52, 88, 96, 120, and 124.

Almost Integers

As Wikipedia puts it, “In recreational mathematics an almost integer is an irrational number that is surprisingly close to an integer.” Some of these Almost Integers seem downright spooky until you understand why they’re so close to an integer.

For example, the powers of phi (aka the golden ratio) get eerily close to integers as the power increases: phi¹¹ is almost exactly 199; phi¹² is almost exactly 322; phi¹³ is almost exactly 521, etc. The reason for this is that as phi is multiplied by itself an increasing number of times, the integral part of the product gets larger, but the irrational part gets smaller because of the way the terms cancel each other out. As the exponent gets larger, the powers of phi approach the Lucas numbers, a Fibonacci-like sequence: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521…

Mathworld has a great list of Almost Integers, including the one immortalized in an XKCD strip.

Mathematical coincidences

Many seeming coincidences encountered in math are not coincidences at all, but rather logical consequences of the properties of the numbers in question. For example, Euler’s famous identity (e^iπ + 1 = 0) is not an amazing coincidence but rather a logical consequence of the laws of math (as well as a thing of beauty.)

However, there are plenty of bona fide coincidences, some of which are discussed here. One particularly interesting coincidence: π⁴ + π⁵ is almost exactly equal to e⁶, within 0.000005%. As far as anyone knows, there’s no mathematical reason for these two numbers to be this close, and yet they are.