An untouchable number is a positive integer that is not the sum of the proper divisors of any number. The first few are 2, 5, 52, 88, 96, 120, 124, 146, ... (OEIS A005114). Erdős has proven that there are infinitely many.

It is thought that 5 is the only odd untouchable number. This would follow from a very slightly stronger version of the Goldbach conjecture, namely the conjecture that every even integer  is the sum of two distinct primes. Suppose  is an odd number greater than 7. Then  by the conjecture, and so the proper divisors of , which are 1, , and , sum to , and so  is not untouchable. 1, 3 and 7 are not untouchable, being the sum of the proper divisors of 2, 4, and 8, respectively. That leaves 5 as the only odd untouchable number (F. Adams-Watters, pers. comm., Aug. 4, 2006).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 840, 1972.

Guy, R. K. "Untouchable Numbers." §B10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 66-67, 1994.

Sloane, N. J. A. Sequence A005114/M1552 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 60, 1986.