2015 AMC 12B Problem 18

Question

For every composite positive integer

n,

define

r(n)

to be the sum of the factors in the prime factorization of

n.

For example,

r(50) = 12

because the prime factorization of

50

is

2 \cdot 5^2,

and

2 + 5 + 5 = 12.

What is the range of the function

r,

\{r(n) : n \text{ is a composite positive integer}\}?

the set of positive integers

the set of composite positive integers

the set of even positive integers

the set of integers greater than

3

the set of integers greater than

4

The smallest value comes from the smallest composite,

4 = 2 \cdot 2

Show every integer greater than

3

is attained, e.g.

r(2^k) = 2k

and

r(2^k\cdot 3) = 2k + 3

A composite number has at least two prime factors (with multiplicity), and the smallest prime is

2,

so the least possible value is

2 + 2 = 4.

Every integer greater than

3

is attained:

r(2^k) = 2k

covers the even values

\ge 4,

and

r(2^k\cdot 3) = 2k + 3

covers the odd values

\ge 5.

So the range is the integers greater than

3.

Thus, the correct answer is D.

Problem 18 in other years