06-21-2026, 11:49 AM
Untouchable number
BY WIKIPEDIA
SUMMARY
An untouchable number is a positive integer that cannot be written as the sum of all the proper divisors of any positive integer; in other words, it is not produced by the aliquot sum function. For example, 4 is not untouchable because it equals the proper divisors of 9: ($1+3=4$), while 5 is untouchable because no number has proper divisors that add up to 5.
The first untouchable numbers are ($2, 5, 52, 88, 96, 120,\dots$). Mathematicians have studied these numbers for centuries, and it is known that infinitely many exist (proved by Paul Erdős). It is believed that 5 is the only odd untouchable number, although this remains unproven and is connected to a stronger form of the Goldbach conjecture.
ARTICLE
BY WIKIPEDIA
SUMMARY
An untouchable number is a positive integer that cannot be written as the sum of all the proper divisors of any positive integer; in other words, it is not produced by the aliquot sum function. For example, 4 is not untouchable because it equals the proper divisors of 9: ($1+3=4$), while 5 is untouchable because no number has proper divisors that add up to 5.
The first untouchable numbers are ($2, 5, 52, 88, 96, 120,\dots$). Mathematicians have studied these numbers for centuries, and it is known that infinitely many exist (proved by Paul Erdős). It is believed that 5 is the only odd untouchable number, although this remains unproven and is connected to a stronger form of the Goldbach conjecture.
ARTICLE
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