Untouchable number

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 can be made up of the sum of the proper divisors of 9, i.e. 1 & 3. The number 5 is untouchable as a similar thing cannot be done.

The first fifty-three untouchable numbers are (sequence A005114 in OEIS):

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, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658

5 is believed to be the only odd untouchable number, but this has not been proven: it would follow from the truth of the Goldbach conjecture. Thus it appears that besides 2 and 5, all untouchable numbers are composite numbers. No perfect number is untouchable, since, at the very least, they can be expressed as the sum of their own proper divisors.

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.

Term a(n) in Sloane's A070015 gives the smallest number whose proper divisors add up to n, but zeros for the untouchable numbers.

See also

• Perfect number
• Prime number
• Composite number

External links

• Adams-Watters, Frank and Weisstein, Eric W., "Untouchable Number" from MathWorld.

References

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