The term "weird number" also refers to a phenomenon in two's complement arithmetic.

In mathematics, a weird number is a natural number that is abundant but not semiperfect. ^[1] In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... A powerful number is a positive integer m that for every prime number p dividing m, p2 also divides m. ... In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. ... An Achilles number is a number that is powerful but not a perfect power. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, an almost perfect number (sometimes also called slightly defective number) is a natural number n such that the sum of all divisors of n (the divisor function &#963;(n)) is equal to 2n _ 1. ... In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function &#963;(n)) is equal to 2n + 1. ... In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. ... In mathematics, a k-hyperperfect number (sometimes just called hyperperfect number) is a natural number n for which the equality n = 1 + k(&#963;(n) &#8722; n &#8722; 1) holds, where &#963;(n) is the divisor function (i. ... A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. ... In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. ... In mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a natural number that has no semiperfect proper divisor. ... A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... In mathematics, a highly abundant number is a certain kind of natural number. ... In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. ... In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. ... A highly composite number is a positive integer which has more divisors than any positive integer below it. ... In mathematics, a superior highly composite number is a certain kind of natural number. ... In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ... In mathematics a solitary number is number which does not have any friends. Two numbers m and n are friends if and only if σ(m)/m = σ(n)/n. ... In mathematics, a sublime number is a positive integer which has a perfect number of positive divisors (including itself), and whose positive divisors add up to another perfect number. ... A harmonic divisor number, or Ore number, is a number whose divisors, averaged in a harmonic mean, results in an integer. ... A frugal number is a natural number that has more digits than the number of digits in its prime factorization (including exponents). ... An equidigital number is a number that has the same number of digits as the number of digits in its prime factorization (including exponents). ... An extravagant number (also known as a wasteful number) is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents). ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... ... The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... “Superset” redirects here. ...

The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2+4+6 = 12.

The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ... (sequence A006037 in OEIS). It has been shown that an infinite number of weird numbers exist, and the sequence of weird numbers has been proven to have positive asymptotic density. ^[2] The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... In mathematics, a sequence a1, a2, ... , an, with the aj positive integers and aj < aj+1 for all j, has natural density (or asymptotic density) α, where 0 ≤ α ≤ 1, if the proportion of natural numbers included as some aj is asymptotic to α. More formally, if we define the counting function A...

It is not known if any odd weird numbers exist; if any do, they must be greater than 10⁶.^[2]

References

External link

• Eric W. Weisstein, Weird number at MathWorld.