The abundancy ratio of a number is defined as the sum of the number's divisors divided by the number itself.

An alternative definition is the sum of the reciprocals of the divisors.

(1) Using the first definition, we want a number with a small abundancy ratio. We recognize that the ratio must be greater than 1 as all integers greater than 1 have at least two divisors; 1 and the number.

It seems likely that primes have small abundancy ratios. The abundancy ratio for a prime is `(1+n)/n` .

Then `(1+n)/n<1.001==>n>1000` . So we search for the smallest prime greater than 1000 and find 1009.

A(1009)=`1010/1009~~1.00099108<1.001`

**So 1009 works, as do all primes greater than 1009.**

(2) **Any power of 2 greater than 8 works. See link below**

e.g. `A(16)=31/16=1.9375`

(3) We use the alternative definition. If n is a multiple of 6 then it has 1,2,3,6 as factors, at least. Then the sum of the reciprocals of the factors is `1/2+1/2+1/3+1/6=2` at least. (If the number is greater than 6, then there are other factors and the sum of the reciprocals of the divisors is greater.

**Further Reading**