Recently I asked this question and quickly got back some excellent responses. I asked the question because I came across a paper by Eric Rowland called "A Natural Prime-Generating Recurrence". The main results on page 4. It appears there was much fanfare in academic news when the paper was released which turns out to be basically the question I asked yesterday except, instead of subtracting the lpf, he added it causing his function to climb indefinitely, whereas mine is decreasing.
So basically Rowland's iterations is $$f(n) = n + lpf – 1$$ starting at 5 (where lpf = least prime factor of $n$) which yields: $$5→9→11→21→23→45→47→93→95→99→101→201→203→209…$$
The sequence above is exactly the sequence Roweland has in column 2 on page 4 of his article without the duplicates.
Am I missing why Roweland's sequence is a significant discovery?
This is OEIS sequence A190894, which is not Rowland's sequence A106108 but an auxiliary sequence used in proving the properties of Rowland's sequence. Rowland's sequence is $$ 7, 8, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69, \ldots $$ which has the recurrence $a(n) = a(n-1) + \gcd(n, a(n-1))$ with $a(1) = 7$, and the property that the difference between any two terms is either $1$ or a prime.