The hardest function in math. What's the most 1s an n-state Turing machine can write before halting?
Σ(n) = max 1s written by any halting n-state Turing machine on a blank tape.
Known values: Σ(1)=1, Σ(2)=4, Σ(3)=6, Σ(4)=13.
For 5 states, the champion writes 4098 ones in 47,176,870 steps.
Σ(n) grows faster than any computable function — it's literally uncomputable.
Pick a champion machine below and watch it run step by step.