An untouchable number is 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 ...

that cannot be expressed as the sum of all 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 of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the

aliquot sum In number theory, the aliquot sum ''s''(''n'') of a positive integer ''n'' is the sum of all proper divisors of ''n'', that is, all divisors of ''n'' other than ''n'' itself. That is, :s(n)=\sum\nolimits_d. It can be used to characterize the prim ...

function. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable.

Examples

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, 52, 88, 96, 120, 124,

146 146 may refer to: *146 (number), a natural number *AD 146, a year in the 2nd century AD *146 BC, a year in the 2nd century BC *146 (Antrim Artillery) Corps Engineer Regiment, Royal Engineers See also

* List of highways numbered 146 * {{Numbe ...

, 162, 188, 206, 210, 216,

238 __NOTOC__ Year 238 ( CCXXXVIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Pius and Pontianus (or, less frequently, year 991 ''Ab ...

246 __NOTOC__ Year 246 ( CCXLVI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar, the 246th Year of the Common Era ( CE) and Anno Domini ( AD) designations, the 246th year of the 1st millennium, th ...

248 __NOTOC__ Year 248 ( CCXLVIII) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Philippus and Severus (or, less frequently, year 1001 '' ...

262 __NOTOC__ Year 262 (Roman numerals, CCLXII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Gallienus and Faustianus (or, less fre ...

, 268,

276 __NOTOC__ Year 276 ( CCLXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Tacitus and Aemilianus (or, less frequently, year 1029 ...

288 Year 288 ( CCLXXXVIII) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Maximian and Ianuarianus (or, less frequently, year 1041 ...

, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ...

Properties

The number 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, since the sum of the proper divisors of ''pq'' (with ''p'', ''q'' distinct primes) is 1+''p''+''q''. Thus, if a number ''n'' can be written as a sum of two distinct primes, then ''n''+1 is not an untouchable number. It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 7 is an untouchable number, and

1=\sigma(2)-2

3=\sigma(4)-4

7=\sigma(8)-8

, so only 5 can be an odd untouchable number. Thus it appears that besides 2 and 5, all untouchable numbers are

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...

s (since except 2, all even numbers are composite). No

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...

is untouchable, since, at the very least, it can be expressed as the sum of its own 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 (this situation happens at the case for 28). Similarly, none of the

amicable number Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive di ...

s or

sociable number In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and ...

s are untouchable. Also, none of the

Mersenne numbers In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...

are untouchable, since M_n=2ⁿ-1 can be expressed as 2ⁿ's proper divisors' sum. No untouchable number is one more than a

prime number 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 ...

, since if ''p'' is prime, then the sum of the proper divisors of ''p''² is ''p'' + 1. Also, no untouchable number is three more than a prime number, except 5, since if ''p'' is an odd prime then the sum of the proper divisors of ''2p'' is ''p'' + 3.

Infinitude

There are infinitely many untouchable numbers, a fact that was proven by

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

. According to Chen & Zhao, their

natural 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 de ...

is at least d > 0.06.Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen 78:2 (2011), pp. 439-442.

References

* Richard K. Guy, ''Unsolved Problems in Number Theory'' (3rd ed),

Springer Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, 2004 ; section B10.

External links

* {{Divisor classes Arithmetic dynamics Divisor function Integer sequences