number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, a weird number is a

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

that is abundant but not semiperfect. In other words, the sum of the

proper divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s (

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s including 1 but not itself) of the number is greater than the number, but no

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

of those divisors sums to the number itself.

Examples

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 several weird numbers are : 70,

836 __NOTOC__ Year 836 ( DCCCXXXVI) was a leap year starting on Saturday of the Julian calendar, the 836th year of the Common Era (CE) and Anno Domini (AD) designations, the 836th year of the 1st millennium, the 36th year of the 9th century, and th ...

, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... .

Properties

Infinitely many weird numbers exist. For example, 70''p'' is weird for all primes ''p'' ≥ 149. In fact, the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of weird numbers has positive

asymptotic density In number theory, natural density, also referred to as asymptotic density or arithmetic density, is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desi ...

. It is not known if any odd weird numbers exist. If so, they must be greater than 10²¹. Sidney Kravitz has shown that for ''k'' a positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

, ''Q'' a prime exceeding 2^''k'', and :

R = \frac

also prime and greater than 2^''k'', then :

n = 2^QR

is a weird number. With this formula, he found the large weird number :

n=2^\cdot(2^-1)\cdot153722867280912929\ \approx\ 2\cdot10^.

Primitive weird numbers

A property of weird numbers is that if ''n'' is weird, and ''p'' is a prime greater than the

sum of divisors In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...

σ(''n''), then ''pn'' is also weird. This leads to the definition of ''primitive weird numbers'': weird numbers that are not a multiple of other weird numbers . Among the 1765 weird numbers less than one million, there are 24 primitive weird numbers. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form

2^k p q

are primitive, but the existence of infinitely many ''k'' and ''Q'' which yield a prime ''R'' is not guaranteed. It is

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d that there exist infinitely many primitive weird numbers, and

Melfi Melfi ( Lucano: ) is a town and ''comune'' in the Vulture area of the province of Potenza, in the Southern Italian region of Basilicata. Geographically, it is midway between Naples and Bari. In 2015 it had a population of 17,768. Geography On a ...

has shown that the infinitude of primitive weird numbers is a consequence of

Cramér's conjecture In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and ...

. Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.

References

External links

* {{DEFAULTSORT:Weird Number Divisor function Integer sequences

Examples

Properties

Primitive weird numbers

See also

References

External links