(i) The only integers with an odd number of factors are square numbers, because the square root is left over when you pair up the other factors.

So we can investigate the number of factors of the square numbers, to find the smallest one that has 7 factors

1 has 1 factor

4 has 3 factors

9 has 3 factors

16 has 5 factors

25 has 3 factors

36 has 9 factors

49 has 3 factors

64 has 7 factors

To find the smallest integer that has 8 factors needs a bit more thought!

We can exclude prime numbers, as they have exactly 2 factors.

So let’s look at …

6 has 4 factors (1, 2, 3, 6)

8 has 4 factors (1, 2, 4, 8)

10 has 4 factors (1, 2, 5, 10)

12 has 6 factors (1, 2, 3, 4, 6, 12)

14 has 4 factors (1,2, 7, 14)

…No luck so far, but you may have noticed that integers of the form pq, where p and q are both primes, must have 4 factors: 1, p, q and pq. So let’s not bother with them any more!

18 has 6 factors (1, 2, 3, 6, 9, 18)

20 has 6 factors (1, 2, 4, 5, 10, 20)

24 has 8 factors (1, 2, 3, 4, 6, 8, 12, 24)

(ii) The terms in the sequence are the smallest numbers that have 1, 2, 3, 4,…. factors, respectively. So, using this, the 12^th term must be the smallest integer with exactly 12 factors and the 14^th term must be the smallest integer with exactly 14 factors.

For this part you may find it useful to know that there is a way to calculate the number of factors that a integer has, which depends on its prime factors.

For example, since 72 = 2³3² , we can construct factors in the form 2^a3^b , where a could be 0, 1, 2, or 3 (4 choices), and b could be 0, 1 or 2 (3 choices). Therefore there are 4 x 3 = 12 factors of 72.

You may be able to write a program that calculates the number of factors for each integer and then picks out the smallest integer for each number of factors.

We used the formula and also did a bit of trial and improvement!

We think that the smallest number with 12 factors is 60 (1, 2, 3, 4, 5, 6, 10, 12, 15, 30, 60) and that the smallest number with 14 factors is 192 (1, 2, 3, 4, 6, 8, 12, 16, 24, 48, 64, 96, 192).

Challenge 334: Look for the Least