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Untouchable Number


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 n>6 is the sum of two distinct primes. Suppose 2n+1 is an odd number greater than 7. Then 2n=p+q by the conjecture, and so the proper divisors of pq, which are 1, p, and q, sum to 1+p+q=2n+1, and so 2n+1 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).


See also

Goldbach Conjecture

Portions of this entry contributed by Frank Adams-Watters

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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.

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Untouchable Number

Cite this as:

Adams-Watters, Frank and Weisstein, Eric W. "Untouchable Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UntouchableNumber.html

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