Favorite mathematical sequences
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 of a pair of consecutive integers that has a square factor (8 is divisble by 4, a square, and 9 is of course the square of 3).
The third number in the sequence is 48, which is the start of three consecutive integers divisible by a square: 48 is divisible by a square (4), 49 is a square, and 50 is divisible by a square (25).
Next in the sequence is 242, the first of four consecutive NSF (non-square-free) integers: 242 is divisible by 121 (11 squared); 243 is divible by 9; 244 is divisible by 4; and 245 is divisible by 49. The sequence continues with the first of five consecutive NSF integers, the first of six, etc., up to the first of eighteen consecutive NSF integers, which seems to be the highest run found so far. (There is a program available for finding more, but I haven’t tried it out yet.)
The OEIS has another version of this list, with a different number in the tenth position. This is because the smallest consecutive NSF integer run containing exactly ten integers is actually larger than the smallest containing eleven. The first OEIS entry therefore shows the smallest 11-integer run in both the tenth and eleventh spots, whereas the second entry shows the first run of exactly ten integers in the tenth spot.
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.
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...
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 (eiπ + 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.