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An untouchable number is an integer that can not be expressed as the sum of the proper divisors of any integer. The first few 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. 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.
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