After this year, I can always divide my life into triangles

Today is my 33rd birthday. In honor of that, here are some interesting properties of 33.

One from Wikipedia’s list which I like because I have a soft spot for integer partition problems, is that it’s the largest positive integer that cannot be expressed as a sum of different triangular numbers. The others are 2, 5, 8, 12, and 23: see OEIS A053614. There’s an almost-proof of this fact in this compilation of problems from mathematical olympiad selection tests; that compliation cites this review paper of Erdos and Graham on results in combinatorial number theory, but I can’t find the result there! If I make it to 128, it’s the largest number not the sum of distinct squares.

An idea of the proof is as follows: check by enumeration that 34 through 66 can be written as the sum of distinct triangular numbers, where 66 is not used: 34 = 28 + 6, 35 = 28 + 6 + 1, 36 = 36, 37 = 36 + 1, 38 = 28 + 10, …, 66 = 55 + 10 + 1. Then add 66 to each of these to get a way of expressing 67, 68, …, 132 as a sum of distinct triangular numbers – for example 104 = 66 + 38 = 66 + 28 + 10. Add the largest triangular number less than 132 (this turns out to be 120) to each of those decompositions to write each of 133, …, 252 as such a sum. And so on.

Why is this worth singling out from the list? Many of the others include some arbitrary constant, such as:

• “the sum of the first four positive factorials”
• “the smallest odd repdigit that is not a prime number” (a “repdigit” is a number that consists of the same digit repeated, so the constant 10 is hiding here; inf act you could argue this is basically a strange way of stating the identity 33 = 3(10+1))

It’s also pretty cool that 33 is a Blum integer – that is, a product of two distinct primes, each of which is congruent to 3 mod 4. (But it’s not the first Blum integer – that’s 21.)

Another property of 33, which is less negative, is that it’s the first member of the first cluster of three semiprimes (33 = 3 x 11, 34 = 2 x 17, 35 = 5 x 7). That is, it’s the first member of this sequence. In OEIS terms, I’d say that being the first member of a sequence, or the last member of a sequence, is more interesting than being just out in the middle of the sequence somewhere.

The semiprime thing appears to have an arbitrary constant of 3. But there are no clusters of four or more consecutive semiprimes – out of four consecutive integers, one is divisible by 4 – so 33 is the first member of the first cluster of semiprimes of maximal length.

Want to know what’s interesting about some number? You could trawl the OEIS or Wikipedia, or you could go to Erich Friedman’s list, which is a bit more selective, only listing one property of each number. In fact both of my interesting properties of 33 appear here – the semiprime one is, for Friedman, a property of 34, “the smallest number with the property that it and its neighbors have the same number of divisors”.