Untouchable number
Encyclopedia
An untouchable number is a positive integer
that cannot be expressed as the sum
of all the proper divisors of any positive integer (including the untouchable number itself).
For example, the number 4 is not untouchable as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4. The number 5 is untouchable as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2).
The first few untouchable numbers are:
5 is believed to be the only odd untouchable number, but this has not been proven: it would follow from a slightly stronger version of the Goldbach conjecture. Thus it appears that besides 2 and 5, all untouchable numbers are composite number
s. No perfect number
is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisor
s.
There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős
.
No untouchable number is one more than a prime number
, since if p is prime, then the sum of the proper divisors of p2 is p + 1. Also, no untouchable number is three more than a prime number, except 5, since if p is prime (except two) then the sum of the proper divisors of 2p is p + 3.
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
that cannot be expressed as the sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of all the proper divisors of any positive integer (including the untouchable number itself).
For example, the number 4 is not untouchable as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4. The number 5 is untouchable as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2).
The first few untouchable numbers are:
- 2, 5, 5252 (number)52 is the natural number following 51 and preceding 53.-In mathematics:Fifty-two is the 6th Bell number and a decagonal number...
, 8888 (number)88 is the natural number following 87 and preceding 89.- In mathematics :Eighty-Eight is a refactorable number, a primitive semiperfect number and an untouchable number...
, 9696 (number)96 is the natural number following 95 and preceding 97.-In mathematics:Ninety-six is an octagonal number, a refactorable number and an untouchable number. Since it is a multiple of 6, it is a semiperfect number...
, 120120 (number)120 is the natural number following 119 and preceding 121. 120 was known as "the great hundred", especially prior to the year 1700, from the Teutonic Hundert which equalled 120. The number 100, now known commonly as "one hundred" was then known as "the small hundred". It is also known as...
, 124124 (number)124 is the natural number following 123 and preceding 125.-In mathematics:124 is the sum of eight consecutive primes...
, 146146 (number)146 is the natural number following 145 and preceding 147.-In mathematics:146 is an octahedral number as well as a composite number....
, 162162 (number)162 is an even natural number between 161 and 163.-In mathematics:162 is divisible by 2, 3, 6, 9, 18, 27, 54, and 81. But there is no positive integer such that its divisors add up to 162, hence 162 is an untouchable number....
, 188188 (number)188 is the natural number following 187 and preceding 189.-In mathematics:* 188 is an even number* 188 has 6 divisors: 1, 2, 4, 47, 94, 188* 188 is a composite number* 188 is a deficient number, as 148 is less than 188* 188 is a happy number...
, 206, 210210 (number)210 is the natural number following 209 and preceding 211.- In mathematics :210 is a composite number, an abundant number, and the product of the first four prime numbers , and thus a primorial...
, 216216 (number)216 is the natural number following 215 and preceding 217.Since 216 = 3^3 + 4^3 + 5^3 = 6^3, it is the smallest cube that's also the sum of three cubes . It is also the sum of a twin prime...
, 238, 246, 248, 262, 268, 276276 (number)Two hundred seventy-six is the natural number following two hundred seventy-five and preceding two hundred seventy-seven.-Mathematical properties:276 = 15 + 25 + 35...
, 288, 290290 (number)290 is the natural number after 289 and before 291.-In mathematics:The product of three primes, 290 is a sphenic number, and the sum of four consecutive primes . The sum of the squares of the divisors of 17 is 290...
, …
5 is believed to be the only odd untouchable number, but this has not been proven: it would follow from a slightly stronger version of the Goldbach conjecture. Thus it appears that besides 2 and 5, all untouchable numbers are composite number
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
s. No perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
s.
There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
.
No untouchable number is one more than a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, since if p is prime, then the sum of the proper divisors of p2 is p + 1. Also, no untouchable number is three more than a prime number, except 5, since if p is prime (except two) then the sum of the proper divisors of 2p is p + 3.