The term numerology literally translated means simply "the study of numbers", but unfortunately it has been misapplied to the use of numbers in fortune telling, for which the proper term should be numeromancy. It is for this reason that mathematicians tend to use clumsier phrases such as "theory of numbers" or "higher arithmetic" for this branch of study. I have for a long time sought to reinstate the term to its proper usage.
Thus numerology, as I understand the term here, is concerned with the properties of numbers per se. Properties depending on the base of notation, or otherwise using particular numbers chosen for no special reason, are excluded from the exercise. The study of the properties of the expressions for numbers in particular systems of numeration I call digitology.
It is easy to define numbers that are the first of their kind, since any set of cardinal numbers has a first element (this is known as "the well-ordering principle"). Some numbers however are the last of their kind, and others are the only ones of their kind. Can readers provide further examples to add to the following list?
0 — The only number which added to any number does not alter it: n + 0 = n.
1 — The only number which multiplying any number does not alter it: n × 1 = n.
2 — The only number such that any number is a sum of a set of its powers (i.e. no more than one of each). The powers of 2 are 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64, and so on. 0 = +{ }, 3 = +{1, 2}, 5 = +{1, 4}, and so on. This property of 2 is the basis of the binary number system.
7 — The largest prime number for which the Fermat quotient (2^(p-1) - 1)/p is a square. Thus (2^6 - 1)/7 = 9. The only other case is 3 for which (2^2 - 1(/3 = 1. [Source: A. H. Beiler, Recreations in the Theory of Numbers, 1966, p.308.]
24 — The largest number that is divisible by all non-zero numbers less than its square root. That is 24 = 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6. [Source: L. F. Taylor, Numbers, Faber and Faber 1970.]
30 — The largest number with the property that all numbers less than and relatively prime to it are unity or prime. Thus 30 is co-prime with 1, 7, 11, 13, 19, 23, 29. The other, smaller numbers with this property are: 2, 3, 4, 6, 8, 12, 18, 24. [Source: J. V. Uspensky & M. A. Heaslet, Elementary Number Theory 1939.]
70 — The only number whose square is also pyramidal (i.e. the sum of successive squares, beginning with 1). Thus 70^2 = 4900 = 1^2 + 2^2 + 3^2 + ... + 24^2. [Source: Beiler, p.196.]
140 — The largest number whose square is also tetrahedral (i.e. the sum of successive triangular numbers, beginning with 1.) Thus 140^2 = 19600 = 1 + 3 + 6 + 10 + ... + 1176. The only other number with this property is 2 for which 2^2 = 4 = 1 + 3. [Source: Beiler, p.197.]
The above first appeared in The Games and Puzzles Journal vol.1, issue 8+9, p.138, (1988-9).
A PDF version of my study of the mathematics of Figurate Numbers, can be downloaded from the Publications page of the Mayhematics website. This pays particular attention to numbers of circular counters that can be arranged in close-packed patterns to form two or more different geometrical shapes. The following are all the cases less than 1000 (other than the trivial cases of 0 and 1) of numbers that are square, triangular, hexagonal or star-shaped, in two ways; they show all six possible pairings of these four shapes:
Triangular numbers are of the form n.(n+1)/2 and form the sequence: 0, 1, 3, 6, 10, 15. 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, ...
Square numbers are of the form n.n = n^2 and form the sequence: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, ...
Hexagonal numbers are of the form 3.n.(n+1) + 1 and follow the sequence: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, ...
Star numbers are of the form 6.n.(n+1) + 1 and follow the sequence: 1, 13, 37, 73, 121, 181, 253, ...
If we expand the study to include metasquares and diamonds we get further double-patterned numbers:
Metasquare numbers (also called double triangular numbers or pronics) are of the form n.(n+1) and follow the sequence: 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, ...
Diamond numbers, or Diagonal square numbers, are of the form n^2 + (n+1)^2, i.e. the sum of two successive squares and follow the sequence: 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ...