- Untouchable number
An untouchable number is a positive
integer that cannot be expressed as thesum of all theproper divisor s 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 OEIS|id=A005114:
: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 arecomposite number s. Noperfect number is untouchable, since, at the very least, they can be expressed as the sum of their own properdivisor s.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 "p"2 is "p" + 1.Term "a"("n") in Sloane's OEIS2C|id=A070015 gives the smallest number whose proper divisors add up to "n", but zeros for the untouchable numbers.
ee also
*
Perfect number
*Prime number
*Composite number External links
*MathWorld|urlname=UntouchableNumber|title=Untouchable Number|author=Adams-Watters, Frank and Weisstein, Eric W.
References
*
Richard K. Guy , "Unsolved Problems in Number Theory " (3rd ed),Springer Verlag , 2004 ISBN 0-387-20860-7; section B10.
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