to the sum of the proper divisors of 2n. No untouchable number is one more than a prime number, since if p is prime, then the sum of the proper divisors of p2 is p + 1. Also, no untouchable number is three more than a prime number, except 5, since if p is an odd prime then the sum of the proper divisors of 2p is p + 3. Infinitude There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős. According to Chen & Zhao, their natural density is at least d > 0.06. See also Aliquot sequence Nontotient Noncototient Weird number References Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ; section B10. External links Arithmetic dynamics Divisor function Integer sequences