Numbers Analysis Without Haste
Resources: Numbers Aplenty, OEIS
Integer: A whole number. Includes positive numbers, 0, and negative numbers.

Natural Number: A whole, positive number. Some definitions include 0; for simplicity, this page does not.
All natural numbers greater than 1 can be divided into primes and composites

Prime: A number whose only proper divisor is 1.

Composite: A number that has proper divisors other than 1.
Every composite number is the product of a unique series of prime numbers.
Divisor: X is a divisor of N if X evenly divides N.
For example, the divisors of 6 are 1, 2, 3, and 6.

Proper Divisor: X is a proper divisor of N if
X evenly divides N
X does not equal N

For example, the proper divisors of 6 are 1, 2, and 3.

Prime Factors: The unique series of prime numbers that produce a composite number. The elements of the series do not have to be unique.
For example, the prime factors of 24 are 2 x 2 x 2 x 3 = 23 + 31.
[Prime sequences]
Chen Prime: A prime number N where N+2 is also a prime number or is a semiprime number.

Emirp Prime: A prime number that, when reversed, is still prime. Emirp primes exclude palindromes, so the reversed prime must be different than the original prime.

Good Prime: A prime number N such that N2 is greater than the product of each pair of equi-distant primes around it.
For example, consider the first nine primes: 2, 3, 5, 7, 11, 13, 17, 19, 23.
11 is a good prime because 11² = 121 which is greater than each of these products:
(7 x 13 = 91) (5 x 17 = 85) (3 x 19 = 57) (2 x 23 = 46)

Honaker Prime: A prime number N which is the Ith prime number, and SoD(N) = SoD(I).
For example, 131 is a Honaker prime because it is the 32nd prime and 3+2=1+3+1.

Mersenne Prime: A prime number N where N can be written as 2P - 1 for some prime number P.

Ormiston Pair: A pair of consecutive primes which share the same digits.
For example, 18379 and 18397 are an Ormiston pair.

Pierpont Prime: A prime number the is the result of some equation 1 + 2u x 3v, where u and v non-negative.
For example, 2 is a Pierpont prime because 2 = 1 + 20 x 30
For example, 13 is a Pierpont prime because 13 = 1 + 22 x 31

Sophie Germain Primes: A prime number N where 2N+1 is also prime.

Twin Primes: A pair of prime numbers that have a difference of 2.
MECE: Mutually Exclusive and Collectively Exhaustive: every element of a set belongs in one and only one subset.
(Pronounced "meese" like "geese")
MECE over the prime sequence:

Strong Prime: A prime number that is greater than the average of the two surrounding primes.

Balanced Prime: A prime number that is equal to the average of the two surrounding primes. In other words, it is equi-distant between them.

Weak Prime: A prime number that is less than the average of the two surrounding primes.
[Composite sequences]
Semi Prime: A composite number that is the product of two primes.
Also known as biprimes.

Sphenic: A composite number that is the product of three distinct primes.
Brilliant: A composite number that is the product of two primes of the same magnitude (or length).
All brilliant numbers are also semi prime numbers.
Interprime: A composite number that is the average of two consecutive primes numbers. In other words, it is equidistant between two prime numbers.
Repunit: A number that contains all the same digit, such as 111.
RK refers to a number K-digits long made up of just 1's.
Achilles: A powerful number that is not also a perfect power.

Deceptive: A composite M-digit number that is a divisor of the repunit RM-1. These numbers are called "deceptive" because this is usually just an attribute of prime numbers.

Droll: A number whose even prime factors sum to the same total as their odd prime factors.
For example, 72 is a droll number because its prime factors are 23 + 32 and 2 + 2 + 2 = 3 + 3.

Duffinian: A number that is coprime to the sum of its divisors.
For example, 35 is a Duffinian number because it is coprime to 48, the sum of 35's divisors 1, 5, 7, and 35.

Enlightened: A number that begins with concatenation of its distinct prime factors.
For example, 23328 is an enlightened number because its prime factors are 25 + 36 and it starts with the digits "23".

Gapful: A number (of at least three digits) that is divisible by its first digit concatenated to its last digit.
For example, 176 is a gapful number because it is divisible by 16.

Highly Composite: A number that has more divisors than any smaller number.
For numbers less than 7560, all highly composite numbers are also super abundant numbers.

Jordan-Polya: A number that can be written as the product of factorial numbers.

O'Halloran: A number N that cannot be written as 2(AB+BC+CA). In other words, there is no cuboid of size AxBxC with a surface area of N.
There are only 16 such numbers.

Perfect Power: A composite number N that can be written as MK, where M and K are both greater than 1.

Poulet: A composite number N such that 2N-2 is divisible by N. This is noteworthy because this is usually a feature of prime numbers.
Also called Sarrus numbers.
Also called Fermat pseudoprime numbers.

Powerful: A composite number (or 1) that, if it is divisible by prime P, is also divisible by P2.

Practical: A composite number (or 1) N such that every smaller natural number can be written as the sum of proper divisors of N.
For example, 12 is a practical number because all numbers 1 to 11 can be written as the sum of 12's unique proper divisors 1,2,3,4,6.
1 = 1 and so for all the proper divisors.
5 = 2 + 3
7 = 3 + 4
8 = 2 + 6
9 = 3 + 6
10 = 4 + 6
11 = 2 + 3 + 6
All powers of 2 are also practical numbers.
All even perfect numbers are also practical numbers.
Also known as panarithmic numbers.
Named by A. K. Srinivasan who was proving the usefulness of traditional units of measure.

Ruth-Aaron Pair: A pair of consecutive numbers that have the same sum of prime factors. Some definitions count in repeated prime factors, and some do not.

Unprimeable: A composite number N which cannot be turned into a prime number by altering any single digit.
Sum of Digits: SoD(N)
For example, SoD(12) = 1 + 2 = 3

Hoax: A number N such that SoD(N) = SoD(distinct prime factors of N).
For example, 2924 is a hoax number because its prime factorization is 22 x 17 x 43 and 2 + 9 + 2 + 4 = 2 + 1 + 7 + 4 + 3.

Joke: A number N such that SoD(N) = SoD(prime factors of N).
For example, 1776 is a joke number because its prime factorization is 24 x 3 x 37 and 1 + 7 + 7 + 6 = 2 + 2 + 2 + 2 + 3 + 3 + 7.
Also known as the Smith numbers.
MECE over the composite sequence:

Economical: A number N where the count of digits in its prime factorization (not counting exponents of 1) is less than or equal to the count of digits in N.
For example, 121 is an economical number because both it and its prime factorization, 112, contain 3 digits.
The economical sequence is the union of the frugal and equidigital sequences.

Wasteful: A number N where the count of digits in its prime factorization (not counting exponents of 1) is greater than the count of digits in N.
Frugal: A number N where the count of digits in its prime factorization (not counting exponents of 1) is less than the count of digits in N.

Equidigital: A number N where the count of digits in its prime factorization (not counting exponents of 1) is equal to the count of digits in N.
[Sequences including both primes and composites]
Sum of Divisors: SD(N)
For example, SD(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28
Usually written as σ(N), pronounced "sigma of N".

Sum of Proper Divisors: aka Aliquot Sum: SPD(N)
For example, SPD(12) = 1 + 2 + 3 + 4 + 6 = 16

Abundancy Index: The ratio SD(N)/N.
MECE over the natural number sequence:

Abundant: A number N such that SPD(N) > N.
All multiples of an abundant number are also abundant.

Perfect: A number N such that SPD(N) = N.
All even perfect numbers can be written as (2P - 1) * 2P-1
If there are any odd perfect numbers, they must be greater than 101500
All perfect numbers are also triangular and hexagonal numbers.

Deficient: A number N such that SPD(N) < N.
All primes are also deficient, because their only proper divisor is 1.
All proper divisors of a deficient number are also deficient.
Super Abundant: A number that has an Abundancy Index larger than any smaller number.
All super abundant numbers are also abundant numbers.
Admirable: A number N such that SPD(N) minus one proper divisor equals N.
For example, 12 is an admirable number because 12 = 1 + 2 + 3 - 4 + 6.
All admirable numbers are also abundant numbers.
Semiperfect: A number N that is equal to the sum of some of its proper divisors.
Also known as pseudoperfect numbers.
All perfect numbers are also counted as semiperfect numbers.
An abundant number which is not also a pseudoperfect number is called a weird number.

Almost Perfect: A number N such that SD(N) = 2N - 1.
Also known as least deficient numbers.
Also known as slightly defective numbers.
This sequence is synonymous with the powers of 2: 1, 2, 4, 8, 16, 32...

Anti-Perfect: A number N that is equal to the sum of the reverse of its proper divisors.
Ex: N=244, Proper Divisors=1,2,4,61,122, N = 1 + 2 + 4 + 16 + 221

Quasiperfect: A number N such that SD(N) = 2N + 1.
If any such numbers exist, they must be odd, square numbers greater then 1035 and have at least seven distinct prime factors.
Weird: A number N such that SPD(N) > N, and N also cannot equal any subset of its proper divisors. In other words, weird numbers are abundant but not semiperfect.
Zumkeller: A number whose divisors can be partitioned into two sets that sum to the same result.
For example, 12 is a Zumkeller number because its divisors can be partitioned into (2, 12) and (1, 3, 4, 6) which both sum to 14.
All Zumkeller numbers are also composite numbers.
All Zumkeller numbers are either perfect numbers or abundant numbers.
Many practical numbers are also Zumkeller numbers.
Many factorials N! with N>3 are Zumkeller numbers.
Amenable: A number N such that there is a set of N integers that sum to N and multiply to N.
For example, 8 is an amenable number because the eight-element set {-1, -1, 1, 1, 1, 1, 2, 4} sums to 8 and multiplies to 8.

Amicable Pair: A pair of numbers N and M such that SPD(N) = M and SPD(M) = N.

Astonishing: A number N which can be split into two numbers A and B such that A + B = N; also included are numbers where N equals the sum of numbers between A and B inclusive.
For example, 429 is an astonishing number because 429 = 4 + 5 + 6 + ... + 27 + 28 + 29.
For example, 190 is an astonishing number because 190 = 0 + 1 + 2 + ... + 17 + 18 + 19.

Automorphic: A number N such that N2 ends with N.
Ex: 3762 = 141376

Betrothed Pair: A pair of numbers N and M such that SPD(N) = M + 1 and SPD(M) = N + 1.

Cullen: The Nth Cullen number is N x 2N + 1.
Cullen numbers are notable for rarely being prime. They are prime for N = 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419,...
All Cullen numbers are also Proth numbers.

Congruent: A number that is the area of a right triangle that has rational sides.

Curzon: A number N where 2N + 1 is a divisor of 2N + 1.

D-numbers: A number N (greater than 3) such that N divides all KN-2 - K, for K greater than 1 and less than N.
For example, 9 is a D-number because 9 divides 27-2, 47-4, 57-5, 77-7, and 87-8.
Also known as 3-Knodel numbers.

de Polignac: An odd number N that cannot be written as 2K + P, for any prime number P.

Esthetic: A number in which all adjacent digits have a difference of 1. Single-digit numbers are excluded.
For example, 4323 is an esthetic number.

Eulerian: A(N,K) is the number of permutations of {1,...,N} that includes exactly K ascents, where one ascent is when element I in the set is greater than element I-1. All answers to A(N,K) for any values of N and K are Eulerian numbers.
For example, A(4,2) = 11 because these permutations of set {1,2,3,4} include exactly two ascents:
With two ascents: {1,2,4,3} {1,3,2,4} {1,3,4,2} {1,4,2,3} {2,1,3,4} {2,3,1,4} {2,3,4,1} {2,4,1,3} {3,1,2,4} {3,4,1,2} {4,1,2,3}
With more or fewer than two ascents: {1,2,3,4} {1,4,3,2} {2,1,4,3} {2,4,3,1} {3,1,4,2} {3,2,1,4} {3,2,4,1} {3,4,2,1} {4,1,3,2} {4,2,1,3} {4,2,3,1} {4,3,1,2} {4,3,2,1}
Therefore 11 is an Eulerian number.
There are multiple equations that result in each Eulerian number.

Factorial: A number N that can be written as K!
Ex: 6 = 3! = 3x2x1
Ex: 24 = 4! = 4x3x2x1

Factorion: A number N which is equal to the sum of the factorials of its digits
Ex: 145 = 1! + 4! + 5!
The only factorions are 1, 2, 145, 40585

Friedman: A number N that is the result of an equation that uses each digit in N, and can include addition, subtraction, multiplication, division, and powers.
For example, 121 is a Friedman number because 121 = 112.
For example, 13125 is a Friedman number because 13125 = 21 x 53+1.

Happy: Let S(N) be the sum of the squares of the digits of N. By applying S(N) repeatedly to the results of S(N), we generate a sequence. If that sequence includes the number 1, then N is a happy number.
For example, S(44) = 42 + 42 = 16 + 16 = 32
S(32) = 32 + 22 = 9 + 4 = 13
S(13) = 12 + 32 = 1 + 9 = 10
S(10) = 12 + 02 = 1 + 0 = 1
Therefore, 44, 32, 13, 10, and 1 are all happy numbers.
All numbers which are not happy will eventually enter the loop 4, 16, 37, 58, 89, 145, 42, 20, 4.

Highly Totient: A number N with more solutions to the equation TOTIENT(X) = N than any smaller integer.

Hogben: The Nth Hogben number equals N2 - N + 1.
If you write 1 in the center of a grid, and count up as you fill in the squares in a spiral, one diagonal through the spiral will be the Hogben numbers.
H(N) is also the maximal number of 1s that a {0,1} NxN matrix can contain and still be invertible.
Also known as central polygonal numbers.

Idoneal: A number that cannot be written as AB + BC + AC, where A > B > C > 0.
For example, 11 is not an idoneal number because 11 = 2 x 3 + 2 x 1 + 3 x 1 (with A = 3, B = 2, C = 1).
Also known as suitable numbers.
Also known as convenient numbers.

Inconsummate: A number that is not result of any K/SoD(K).

Junction: A number that can be written as K + SoD(K) for at least two different Ks.

Self: A number that cannot be written as K + SoD(K) for any K.
Also known as Colombian numbers.

Kaprekar: A number N such that N2 can be partitioned into two numbers A and B, where A + b = N.
For example, 45 is a Kaprekar number because 452 = 2025, and 20 + 25 = 45.

Leyland: A number that can be written as AB + BA, where A and B are greater than 1.

Lonely: A number that is further from any prime than any lower number. If the number itself is prime, its distance to itself does not count.
If you include 0, then 0 is the first lonely number (at two-distant from 2) and 23 is the next lonely number.
If you do not include 0, then 1 is the first lonely number (at one-distant from 2), 5 is the next, and 23 is the third lonely number.

Lucky: The numbers that survive the lucky sieving process.
Let 1 be the first lucky number, and consider only the odd numbers.
Keep the next surviving number, which is 3. Remove all remaining numbers in a 3-divisible position.
Keep the next surviving number, which is 7. Remove all remaining numbers in a 7-divisible position.
And continue like that forever.

Magic: A magic square is a square layout of the numbers 1 to N such that each row and each column sums to the same number. A magic number is that row/column sum. Magic numbers follow the form (M3 + M)/2, where M is the length of the square.

Magnanimous: A number of at least two-digits, such that you can insert a plus sign "+" at any position between the digits, and the result of the equation is always a prime number.

Modest: A number N that can be divided into two parts A and B, such that N / B leaves remainder A.
For example, 2851111 is a modest number because 2851111 / 1111 leaves remainder 285.

Nude: A number that is divisible by each of its digits. Therefore, it contains no 0s.

Lynch-Bell: A number that has distinct digits, and is divisible by each digit.
All Lynch-Bell numbers are also nude numbers.

Zuckerman: A number that is divisible by the product of its digits.
All Zuckerman numbers are also nude numbers

Partition: The number of ways a set of K identical elements can be partitioned.
For example, when K=4 the resulting partition number is 5. These are the unique partitions:
(x,x,x,x) ((x),(x,x,x)) ((x,x),(x,x)) ((x),(x),(x,x)) ((x),(x),(x),(x))

Primeval: Each number N has a count of how many unique prime numbers can be written with N's digits (the digits may be reused between primes). A primeval number has a higher count than any lower number.
For example, 137 is a primeval number because it can be used to write eleven prime numbers: 3, 7, 11, 13, 17, 31, 37, 73, 137, 173, and 317

Proth: A number that can be written as K x 2N + 1, where N is greater than 0 and 2N is greater than K.

Sastry: A number N that, when concatenated to N + 1, results in a square number.

Square-Free: A number that is not divisible by any square number except for 1.
Digitally Powerful: A number N that can be written as the sum of positive powers of its digits (powers 1 or greater).
For example, 3459872 is a digitally powerful number because N = 31 + 46 + 55 + 96 + 83 + 77 + 221.
Also known as D-Powerful numbers.

Handsome: A number N that can be written as the sum of positive powers of its digits (powers 0 or greater).
Goldbach Conjecture: Every even natural number greater than 2 is the sum of two prime numbers, and every even natural number greater than 6 is the sum of two distinct primes.
Proven to at least 4-trillion.
Untouchable: A number N that is not the result of any SPD(K).
If the Goldbach Conjecture is true, then 5 would be the only odd untouchable number, because larger odd numbers N = 1 + M and M = P + Q (for some primes P and Q), meaning N = SPD(PQ) = 1 + P + Q.

Vampire: A number N (2K digits long) equal to XY (each K digits long), and that N has the same digits as X and Y together.
For example, 1260 is a vampire number because 1260 = 21 x 60.
Solutions where both X and Y end in 0 are excluded for being trivial.
The X and Y values are known as "fangs". Many vampire numbers have multiple pairs of fangs.
Katadrome: A number whose digits are in strictly descending order. That also means the number contains no repeated digits.

Metadrome: A number whose digits are in strictly increasing order. That also means the number contains no repeated digits.

Palindrome: A number whose digits are in the same order when the number is reversed.

Straight-Line: A number whose digits form an arithmetic progression.
For example, 7531 is a straight-line number because it is an arithmetic progression with a step of -2.
MECE over the natural numbers:

Polite: A number that can be written as the sum of two consecutive natural numbers.
Almost all natural numbers are also polite numbers.

Impolite: A number that cannot be written as the sum of two consecutive natural numbers.
Niven: A number N such that N is divisible by SoD(N).
If you exclude single-digit numbers, then all Niven numbers are also composite numbers.

Super Niven: A number N such that N is divisible by the sum of every subset of its digits (that isn't just 0s).
For example, 24 is super Niven because it is divisible by 2, 4, and 2+4.
All super Niven numbers are also Niven numbers.

Moran: A number N such that N / SoD(N) results in a prime number.
All Moran numbers are also Niven numbers.
Fibonacci: A number generated by the recurrence:
F1 = 1
F2 = 1
FN = FN-2 + FN-1

Keith: A number with multiple digits (d1...dJ) that is generated by this recurrence:
K1 = d1
K... = ...
KJ = dJ
KN = KN-1 + ... + KN-J
For example, 1104 is a Keith number because the generated sequence is 1, 1, 0, 4, 6, 11, 21, 42, 80, 154, 297, 573, 1104.
Somehow related to Fibodiv numbers.
Also known as repfigit numbers.

Jacobsthal: A number generated by the recurrence:
J1 = 0
J2 = 1
JN = 2 x JN-2 + JN-1

Lucas: A number generated by the recurrence:
L1 = 2
L2 = 1
LN = LN-2 + LN-1

Gilda: A number with multiple digits (d1...dK) that is generated by this recurrence:
G1 = | d1 - ... - dK |
G2 = d1 + ... + dK
GN = GN-2 + GN-1
For example, 152 is a Gilda number because its Gilda recurrence is 6, 8, 14, 22, 36, 58, 94, 152.

Motzkin: A number generated by the recurrence:
M1 = 1
M2 = 1
MN = ( 3(N-1) x MN-2 + (2N+1) x MN-1 ) / (N+2)
In combinatorics, M(N) is the number of ways you can draw non-intersecting chords between N points on a circle. Rotated/reflected designs are allowed.

Perrin: A number generated by the recurrence:
P1 = 3
P2 = 0
P3 = 2
PN = PN-3 + PN-2

Ulam: A number generated by the recurrence:
U1 = 1
U2 = 2
UN = the smallest integer that is the result of one-and-only-one UJ + UI (where J != I)
For example, the first terms are 1, 2, 3, and 4.
5 is not an Ulam number because it can be written as 1+4 and as 2+3.
Coprime: Two numbers are coprime to each other if they share no prime factors.
Also called relatively prime.
Also called mutually prime.

Totient: Totient(N) is the count of natural numbers less than N that are coprime to N.
Often shortened to the Greek symbol "phi", such as φ(N) or ϕ(N).
Cyclic (group theory): A number N such that Totient(N) and N are coprime.
All prime numbers are also cyclic (group theory) numbers.
All cyclic (group theory) numbers are also square-free numbers.

Cyclic: An M-digit number that will always produce an M-digit product when multiplied by any number 1 to M. In addition, the products will all be cyclic permutations of the original (the same digits in the same order, just rotated a bit).
For example, 142857 is a cyclic number because:
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
If leading-zeros are disqualified, then 142857 is the only cyclic number.
MECE over the natural numbers:

Evil: A number which, in its binary format, has an even number of 1s.

Odious: A number which, in its binary format, has an odd number of 1s.
Pernicious: A number which, in its binary format, has a prime number of 1s.
[Figurative sequences]
Triangular: A number that can be written as N(N + 1) / 2. In other words, a number of blocks that can be arranged into a filled triangle. In other words, the sum of all numbers from 1 to K.
For example, 10 is a triangular number because you can arrange 10 blocks into a row of one, of two, of three, and of four, forming a triangle.

Square: A number that can be written as N2. In other words, a number of blocks that can be arranged into a filled square.
For example, 16 is a square number because you can arrange 16 blocks into four rows of four.
All square numbers are also composite numbers.

Star: A number that can be written as 6N(N-1) + 1. In other words, a number of blocks that can be arranged into a filled six-pointed star.
For example, 13 is a star number because you can arrange 13 blocks into rows one, four, three, four, and one.
Star numbers coincide with centered dodecagonal numbers.

Cube: A number that can be written as N3.
All cube numbers are also composite numbers.
Padovan: A number generated by the recurrence:
P0 = 1
P1 = P1 = 0
PN = PN-1 + PN-3
If you make an equilateral triangle with sides the length of the each Padovan number, they form a spirally shape.

Pronic: A number that can be written as N(N + 1). Thus it is always double a triangle number.
Also called oblong numbers.
Also called rectangular numbers.
Also called heteromecic numbers.
Toothpick: The number of toothpicks at each stage of the toothpick diagram.
The toothpick diagram is started with one toothpick. At each iteration, a new toothpick is placed, perpendicularly and centered, at every open end of a toothpick.
[Combinatorial sequences]
Bell: The Nth Bell number is the number of ways unique N elements can be partitioned into non-empty sets.
For example, B(3) = 5 because (a,b,c) can be partitioned into:
((a),(b),(c)), ((a),(b,c)), ((a,b),(c)), ((a,c),(b)), and ((a,b,c))

Cake: The number of pieces a cube (cake) can be divided into using K planar cuts. Cake numbers are of the form (K3 + 5K + 6) / 6.

Catalan: A number that can be written as (1/N)(2N choose N) which is the same as (2N)!/(N!(N+!)!).
C(N) is the number of ways a 2D staircase with N steps up can be made with N rectangles.
C(N) is also the number of ways a regular N-gon can be divided into N-2 triangles, taking into account different orientations as distinct.
Also called Segner numbers.
See Genjimon - a subset of catalan numbers.

Pancake: The number of pieces a circle (pancake) can be divided into using K linear cuts. Pancake numbers are of the form (K2 + K + 2) / 2.
[Sequences That Go Up and Down]
Aliquot Sequence: Starting from any number N, proceed to SPD(N). If the sequence reaches 1, it ends.

(Sociable): Numbers within Aliquot Sequence loops.
Ex: 14288, 15472, 14536, 14264, 12496, repeated

Hailstone Sequence: Starting from any number N
If N is even, proceed to N/2
If N is odd, proceed to 3N + 1
Also known as wondrous numbers.

(Collatz Conjecture): That all hailstone sequences end at 1.
Also known as the 3n + 1 Problem.
Also known as the Ulam Conjecture.
Also known as Kakutani's Problem.
Also known as the Thwaites Conjecture.
Also known as Hasse's Algorithm.
Also known as the Syracuse Problem.
[Permutations of the Natural Numbers]
Permutations of the Natural Numbers: A sequence that includes each natural number exactly once, and is not simply ordered 1,2,3,4,5...
Codigit: Two numbers are codigit if they have no digits in common (like coprime, but with digits).
A term I'm using - can't find a synonym in use.
Codigit Seidov Corneth:
A(1) = 1
A(2) = 2
A(N) = smallest natural number not in the sequence yet that is codigit to A(N-1) and A(N-2).
First terms: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 33, 11, 20, 34, 15, 26, 30, 14, 25, 36, 17, 24, 35, 16, 27, 38, 19, 40, 23, 18, 44, 29, 13, 45, 28, 31, 46, 50, 12, 37, 48, 21, 39, 47, 51, 32, 49, 55, 60, 41, 52, 63, 70, 42, 53, 61, 72, 43, 56, 71, 80, 54, 62, 73, 58, 64, 77, 59, 66, 74, 81, 65, 79
52 is the first non-trivial number that is equal to its index (if I'm counting correctly).

EKG:
A(1) = 1
A(2) = 2
A(N) = smallest natural number not in the sequence yet that shares a factor with A(N-1). 1 does not count as a factor.
First terms: 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 22, 11, 33, 27, 30, 25, 35, 28, 26, 13, 39, 36, 32, 34, 17, 51, 42, 38, 19, 57, 45, 40, 44, 46, 23, 69, 48, 50, 52, 54, 56, 49, 63, 60, 55, 65, 70, 58, 29, 87, 66, 62, 31, 93, 72, 64, 68, 74, 37, 111, 75, 78, 76, 80, 82
Named for its early similarity to an electrocardiogram graph.

Recamán's: non-increasing, and includes all natural numbers exactly once (or at least, no more than once?)
A(0) = 0
A(N) = A(N-1) - N (if result is positive and not in the sequence already)
A(N) = A(N-1) + N (otherwise)
First terms: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155
[Complementary sequences and Non-Increasing sequences]
First Differences: For any sequence A, a complementary sequence can be generated by taking the differences of all terms: BK = AK - AK-1.
Hofstadter: complementary sequences that include all natural numbers exactly once
A(1) = 1
A(N) = A(N-1) + B(N-1)
B(N) = lowest natural number not in A or B yet
B is the first differences of A.
First terms A: 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, 285, 312, 340, 369, 399, 430, 462, 495, 529, 565, 602, 640, 679, 719, 760, 802, 845, 889, 935, 982, 1030, 1079, 1129, 1180, 1232, 1285, 1339, 1394, 1451, 1509, 1568, 1628, 1689
First terms B: 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25
[Other]
Radical of N: The radical of a number N is the product of the distinct prime factors of N.
Fundamental Theorem of Arithmetic: aka Unique Factorization Theorem: Every integer greater than 1 can be represented exactly one way (apart from rearrangements) as the product of one or more primes.

Euclid's Theorem: There are infinitely many primes.
The Hyperoperation Sequence:
Addition: K + N = N + 1 + 1 + 1... until the equation contains K number of 1s
Multiplication: K x N = N + N + N... until the equation contains K number of Ns
Exponentiation: NK = N x N x N.... until the equation contains K number of Ns
Tetration: NK = N^(N^(N...)) until the equation contains K number of Ns