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The only odd untouchable (Posted on 2016-04-12) Difficulty: 3 of 5
An untouchable number is a positive integer that is not the sum of the proper divisors of any number(see OEIS A005114).

It is thought that 5 is the only odd untouchable number.

Why?

No Solution Yet Submitted by Ady TZIDON    
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Solution Goldbach Solution Comment 1 of 1
The Goldbach conjecture states every even number starting with 4 can be written as the sum of two primes.  This is widely believed to be true.  A slightly stronger version states that every even number starting with 8 can be written as the sum of two distinct odd primes.

Assume the stronger version is in fact true.  Then every odd number k starting with 9 can be written as k=1+p+q where p and q are distinct odd primes.  Then let x be the number x=p*q.  Then the sum of the proper divisors of x is k.

This leaves 1, 3, and 7 to check by hand.  1 is touchable by any prime.  3 is touchable by 4 (1+2).  7 is touchable by 8 (1+2+4).  This leaves 5 as the only untouchable odd number.

  Posted by Brian Smith on 2016-04-12 12:37:55
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