A new prime-indexed additive sequence and its properties

Question

I would like to share and ask about the following integer sequence that I have been experimenting with, while looking for "simple but rich" sequences in the spirit of OEIS contributions.

I start from $a(1)=p\;$($p\in P$, that is a prime number) and for $n>1$ I pose $$a(n)=a(n-1)+S_{n-1}$$ where $$S_{n-1}=\sum_{i<n,円:,円a(i)\in P}i$$ If $,円p=2,\;a(1)=2\implies S_1=1\implies a(2)=2+1=3\implies S_2=1+2=3\implies a(2)\in P,\;a(2)|S_2.$ Therefore, $a(n+1)=a(n)+3,円$ for $n\gt1,円$ and the sequence becomes an AP (Arithmetic Progression).

Now, instead, for $,円p=3,円$ we have: $3,ドル 4, 5, 9, 13, 22, 31, 47, 71, 104, 137, 181, 237,\dots$$ and the sequence seems not to reach an AP ...

The same behaviour occurs for each odd prime, that is new primes income regurarly, even if with decreasing frequency. But we are not certain that sooner or later the sequence will not reach the fatal condition $a(L)\in P$ and $a(L)|S_L$ for some index $L$.

Using Dirichlet's theorem, it should not be difficult to prove that the sequence contains only a finite number of prime terms if and only if there exists an index $L$ such that $,円a(L)\in P,円$ and $,円a(L)|S_L,円$.

I think the sequences should grow as $$a(n)\sim\frac{cn^3}{\log n}$$ for some constant $c$.

Open questions:

• can one prove rigorously that for $p\gt2$ the sequence cannot degenerate into an arithmetic progression?
• more strongly: does the sequence contain infinitely many primes for every $p\gt2$?

Many thanks.

• Dermot Craddock: The "should not be difficult" is indeed not difficult. The sequence becomes (at some point) an arithmetic progression if and only if it contains only finitely many primes. [The step size increases at every prime, and only at primes.] Say the last prime is $a(L)$. From that point on, the step size is $S_L$. By Dirichlet's theorem, since the progression $a(L) + k\cdot S_L$ contains only finitely many primes,we have $\gcd(a(L),S_L) > 1$. But since $a(L)$ is a prime that means $a(L) \mid S_L$. The converse is clear.
• Misha Lavrov: A simpler version of this recurrence is used in the definition of oeis.org/A131073.