Untouchable number

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 {{OEIS|id=A005114}}:

Related Topics:
Integer - Sum - Proper divisor

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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

Related Topics:
2 - 5 - 52 - 88 - 96 - 120

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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.

Related Topics:
Composite number - Perfect number - Divisor

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There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős.

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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.

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