Mere coincidence? (prime factors)

Mere coincidence? (prime factors)

Whether some things in mathematics are mere coincidences might keep
philosophers busy for 100,000 aeons, but maybe when such a coincidence
gets exploited then it's not a "mere" coincidence any more.
So time for a somewhat imprecise question: a list of prime factorizations
shows us this: $$ 1445=5\cdot17\cdot17 $$ and if you were doing
factorizations of consecutive numbers one by one, maybe you'd be just a
teensy bit surprised to see $17$ twice in a row; maybe you'd even stir a
bit before descending back into deep sleep. But then the very next number
(if you're going downward) is: $$ 1444=2\cdot2\cdot19\cdot19 $$ Two
squares of somewhat....um....big...primes in a row!
Does this fit into some grander design that the god of mathematics dreamed
up when he wasn't busy distributing zeros of the zeta function as if they
were eigenvalues of random matrices?
After noticing that I noticed this: $$ \sqrt{\frac54} =
1+\cfrac{1}{8+\cfrac{1}{2+\cfrac{1}{8+\cfrac{1}{2+\cfrac{1}{8+\cfrac{1}{2+\ddots}}}}}}
$$ whereas $$ \frac{19}{17} = 1+\cfrac{1}{8+\cfrac{1}{2}} $$