[Guest]

Janice writes a sequence of integers starting with the number 12. Each subsequent

integer she writes is chosen randomly with equal probability from amongst the positive divisors of the previous integer (including the possibility of the integer itself). She keeps writing integers until she writes the integer 1 for the first time, and then she stops.

An example of one such sequence is 12, 6, 6, 3, 3, 3, 1.

What is the expected value of the number of terms in Janice’s sequence?

[unreality]

about 4.03362493

[Guest]

Lets call the expected number of items following the number n, E(n).

E(1) = 0

E(2) = 1 + 0.5*[E(1) + E(2)] = 2

E(3) = 1 + 0.5*[E(1) + E(3)] = 2

E(4) = 1 + 0.33*[E(1) + E(2) + E(4)] = 2.5

E(6) = 1 + 0.25*[E(1) + E(2) + E(3) + E(6)] = 2.67

E(12) = 1 + 0.167*[E(1) + E(2) + E(3) + E(4) + E(6) + E(12)] = 3.033

And the 12 itself gives 4.033... or 121/30

unreality - looks like you've approximated that using a program, buts its very close .