Numbers |
Analysis Without Haste |

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.

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

[Prime sequences]

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]

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, 11^{2}, 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]

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 (2^{P} - 1) * 2^{P-1}

If there are any odd perfect numbers, they must be greater than 10^{1500}

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 10^{35} 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.

Friedman Solutions

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.

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]

[Combinatorial sequences]

[Sequences That Go Up and Down]

[Permutations of the Natural Numbers]

[Complementary sequences and Non-Increasing sequences]

[Other]