Weird number

 ( Divisor Function ) 

In number theory, a weird number is a natural number that is abundant number but not semiperfect.

Section B2. In other words, the sum of the ( including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.

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

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Properties Infinitely many weird numbers exist.

(2025). 9781402042157, Springer-Verlag. ISBN 9781402042157

For example, 70 p is weird for all prime number p ≥ 149. In fact, the set of weird numbers has positive asymptotic density.

It is not known if any odd weird numbers exist. If so, they must be greater than 10²¹. -- comments concerning odd weird numbers

Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2 ^k, and

$R\; =\; \backslash frac\{2^kQ-(Q+1)\}\{(Q+1)-2^k\}$

also prime and greater than 2 ^k, then

$n\; =\; 2^\{k-1\}QR$

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

$n=2^\{56\}\backslash cdot(2^\{61\}-1)\backslash cdot153722867280912929\backslash \; \backslash approx\backslash \; 2\backslash cdot10^\{52\}.$

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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: 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 that there exist infinitely many primitive weird numbers, and Giuseppe Melfi has shown that the infinitude of primitive weird numbers is a consequence of Cramér's conjecture. Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.

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See also

• Untouchable number

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External links

Categories: Divisor Function, Integer Sequences

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