number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

, a weird number is a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

that is abundant but not

semiperfect In number theory, 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. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. ...

. 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 divisible by ...

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 divisible by ...

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

subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subset o ...

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 few weird numbers are : 70,

836 __NOTOC__ Year 836 ( DCCCXXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. Events By place Abbasid Caliphate * Driven by tensions between his favoured Turkish guard and the popul ...

, 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

''p'' ≥ 149. In fact, the set 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 des ...

. 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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

, ''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 σ(''n''), then ''pn'' is also weird. This leads to the definition of ''primitive weird numbers'', i.e. weird numbers that are not a multiple of other weird numbers . There are only 24 primitive weird numbers smaller than a million, compared to 1765 weird numbers up to that limit. 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 (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

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 infiniteness 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 t ...

. 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