Untouchable number

Unsolved problem in mathematics:

Are there any odd untouchable numbers other than 5?

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:

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... (sequence A005114 in the OEIS)

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.^[1] Thus it appears that besides 2 and 5, all untouchable numbers are composite numbers. No perfect number is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisors. Similarly, none of the amicable numbers or sociable numbers are untouchable.

There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős.^[2]

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 an odd prime then the sum of the proper divisors of 2p is p + 3.

References

^ The stronger version is obtained by adding to the Goldbach conjecture the further requirement that the two primes be distinct - see Adams-Watters, Frank and Weisstein, Eric W. "Untouchable Number". MathWorld.{{cite web}}: CS1 maint: multiple names: authors list (link)
^ P. Erdos, Über die Zahlen der Form $\sigma (n)-n$ und $n-\phi (n)$ . Elemente der Math. 28 (1973), 83-86, [1]

Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B10.

External links

Sloane, N. J. A. (ed.). "Sequence A070015 (Least m such that sum of aliquot parts of m equals n or 0 if no such number exists)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[1] The stronger version is obtained by adding to the Goldbach conjecture the further requirement that the two primes be distinct - see Adams-Watters, Frank and Weisstein, Eric W. "Untouchable Number". MathWorld.{{cite web}}: CS1 maint: multiple names: authors list (link)

[2] P. Erdos, Über die Zahlen der Form $\sigma (n)-n$ und $n-\phi (n)$ . Elemente der Math. 28 (1973), 83-86, [1]

[1]

[2]

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Descartes Refactorable Superperfect

See also

References

External links