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user21820
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Recurrent formula for Difference between $n$-th and $(n-1)$-th composite numbers

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Notamathematician
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Recurrent formula for composite numbers

Let $f(n)$ = 1 if $n$ belongs to A014689, $\operatorname{prime}(n)-n$, the number of nonprimes less than $\operatorname{prime}(n)$. Here $\operatorname{prime}(n)$ is the $n$-th prime number, $\operatorname{prime}(1)=2$.

Let $a(n)$ be the $n$-th composite numbers, $a(1)=4$.

Then I conjecture that

$$a(n) = 1 + a(n-1) + f(n)$$

Is there a way to prove it?