This is a simple variant of Eric Angelini's sequence involving chasing base-ten Harshad numbers. Instead of adding, we'll be multiplying...

Let's start with 23. Is 23 divisible by the sum of its digits (5)?

No. Then we multiply 23 by 5 => 115

 

Is 115 divisible by (1+1+5) = 7?

No. Then we multiply 115 by 7 => 805

 

Is 805 divisible by (8+0+5) = 13?

No. Then we multiply 805 by 13 => 10465

Is 10465 divisible by (1+0+4+6+5) = 16?

No. Then we multiply 10465 by 16 => 167440

 

Is 167440 divisible by (1+6+7+4+4+0) = 22?

No. Then we multiply 167440 by 22 => 3683680

Is 3683680 divisible by (3+6+8+3+6+8+0) = 34?

No. Then we multiply 3683680 by 34 => 125245120

Is 125245120 divisible by (1+2+5+2+4+5+1+2+0) = 22?

Yes! We have come to the end and taken six steps to do so.

In fact, 23 is the smallest number that will come to an end in this fashion after six steps. We are accumulating factors with every iteration, so it would be surprising to find numbers that go on for a long time.

The smallest numbers that iterate 0, 1, 2, 3, ... 36 times are: 1, 14, 13, 22, 11, 61, 23, 41, 83, 38, 653, 239, 5047, 1019, 932, 698, 21989, 35161, 17762, 103799, 37861, 36184, 18092, 610979, 115546, 4014896, 1836482, 4645373, 487843, 45490984, 86908406, 49216216, 762759839, 109828346, 519088298, 496001336, 2298452614.

I've put here (for each of these) the factored initial starting number followed by the factorizations of each sum-of-the-digits, until divisibility occurs.