Testing AI

I thought I might challenge AI to guess the next number in a given sequence. This was my query:

Can you propose what the next number in this sequence of numbers might be and explain your reasoning$$1, 3, 7, 13, 7, 31, 43, 19, 73, 13, 37, \dots$$First up I asked Grok and got a rather curious response:

Uh-oh, too much information for me to digest all at once. You know, sometimes less is more!

That wasn't much help so I turned to Gemini and ChatGPT. They both accepted the challenge and tried their best but neither came up with anything convincing. Finally I gave the latter the following clue:

I'll give a clue. The numbers are generated by a simple quadratic function (using the integers 0, 1, 2, 3 , ... progressively as input) that collapses composite numbers when they appear according to a simple division rule. 

 Nice clue — that nails it.

The quadratic is \(f(n)=n^2+n+1\) for integer inputs \(n=0,1,2,\dots \). 

The collapse rule: whenever \(f(n) \) is composite, divide it by its smallest prime factor (and repeat if necessary) until the result is prime.

Checking the given terms (I'll show \(n\) \( \rightarrow \) raw \(f(n)\) \( \rightarrow \) collapsed result):

• \(n=0: f=0^2+0+1=1 \rightarrow 1\)

• \(n=1: f=1^2+1+1=3 \rightarrow 3\)

• \(n=2: f=4+2+1=7 \rightarrow 7 \)

• \(n=3: f=9+3+1=13 \rightarrow13 \)

• \(n=4: f=16+4+1=21 \rightarrow 21/3=7 \)

• \(n=5: f=25+5+1=31 \rightarrow 31\)

• \(n=6: f=36+6+1=43 \rightarrow 43 \)

• \(n=7: f=49+7+1=57 \rightarrow 57/3=19 \)

• \(n=8: f=64+8+1=73 \rightarrow 73\)

• \(n=9: f=81+9+1=91 \rightarrow 91/7=13 \)

• \(n=10: f=100+10+1=111 \rightarrow 111/3=37\)

All match your sequence.

Now the next input:$$n=11: f(11)=121+11+1=133 \rightarrow 133/7 = 19$$ So the next number in the sequence is 19.

So a little clue and ChatGPT, as it said, nailed it. This was just a little experiment to see what how AI would handle a problem that is actually quite challenging. The sequence is NOT listed in the OEIS by the way.

I challenged ChatGPT to come with a similar challenge and this is what it generated:

Could you propose a similar but different challenge. You should try to be creative and innovative but adhering to the general algorithm, namely generate a sequence of numbers according to a definite rule but modify certain numbers according to another definite rule whenever they satisfy a certain preestablished criterion. Go for it.

Nice — here’s a fresh puzzle in the same spirit as yours: a definite generating rule (a simple quadratic) plus a deterministic “collapse” rule that kicks in whenever the raw value meets a criterion.

I had to take a screenshot of the puzzle as copying and pasting introduces tedious formatting challenges.