## 500 Triangles

Posted by Karl Sharman on July 28, 2014 – 7:02 am

A sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

The first 7 terms would be: 1, 3, 6, 10, 15, 21, 28

Let us list the factors of the first seven triangle numbers:

1: 1

3: 1,3

6: 1,2,3,6

10: 1,2,5,10

15: 1,3,5,15

21: 1,3,7,21

28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

July 30th, 2014 at 5:15 am

76576500

July 30th, 2014 at 5:28 am

842161320 is the first one with more than 1000 divisors

July 30th, 2014 at 7:49 am

I concur with jan’s results. But I cheated by using brute force.

I also checked the first million triangle numbers: the first with over 2000 factors

is 49 172 323 200 and with over 3000 factors is 102 774 672 000.

But as to how to do it with brain rather than brawn eludes me.

July 31st, 2014 at 12:33 am

I too have used brute force to get to the same figures as Jan and Chris. Struggling to find any patterns that might give a clue as to a method!

July 31st, 2014 at 5:54 am

Hi Karl. I’ve read (glanced) up on perfect numbers, abundant numbers, highly composite numbers etc. I don’t think that there is a nice way to do this problem. Even the highly composite numbers aren’t easy to find.

At core of the problem (of solving the posted problem) is that the primes just don’t play nice.

I won’t stick my neck out and say there is no a way to do this, but my guts tells me that that is actually the case. i.e. this is a number-crunchie problem.

So Jan, come clean, did you number crunch too (or are you a genius)?

FYI: t = n(n + 1) / 2

n = 12375 => t = 78578600 = 2^2 * 3^2 * 5^3 * 7 * 11 * 13 * 17

12375 = 3^3 * 5^2 * 11

12376 / 2 = 2^2 * 7 * 13 * 17

As you can see, the 5 has a higher power than the 2 and 3. So already not a highly composite number.

n = 41040 => t = 842161320 = 2^3 * 3^3 * 5 * 7 * 11 * 13 * 19 *41

41040 / 2 = 2^3 * 3^3 * 5 * 19

41041 = 7 * 11 * 13 *41