Untouchable number
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An untouchable number is an integer that can not be expressed as the sum of all the proper divisors of any other integer.
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 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. (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 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 a number u was untouchable and u - 1 was prime, then the sum of the proper divisors of (u - 1)2 would be (u - 1) + 1 = u, creating a contradiction.
Term a(n) in Sloane's gives the smallest number whose divisors add up to n, but zeroes for the untouchable numbers.
See also
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