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April 30

The sequence 2, 3, 5, 8, 11, 15, 19, 24, 29, ...

OK, after several days, there is now finally a question here!

Consider the sequence 2, 3, 5, 8, 11, 15, 19, 24, 29, ..., defined as follows: the first term is 2, and all later terms are defined to be the sum of the previous term and the number of prime terms up to and possibly including that term. This means that once a prime term is reached, the number to add to get the next term increases by 1.

Now, does the sequence contain infinitely many prime terms, or equivalently, does the sequence of first differences eventually contain every positive integer? Perhaps, Dirichlet's theorem on arithmetic progressions might help. GeoffreyT2000 (talk) 04:09, 30 April 2020 (UTC)[reply]

If it helps, this is OEIS A131073, and the entry there claims infinitely many prime terms. --{{u|Mark viking}} {Talk} 04:21, 30 April 2020 (UTC)[reply]

Courtesy link: (sequence A131073 in the OEIS). -- ToE 07:50, 30 April 2020 (UTC)[reply]

As the sequence increases its increment only when there is a prime in the sequence, we apply Dirichlet's theorem directly from that point in the sequence (because $a$ is prime, $a$ and $d$ must be coprime). Because there is an infinite number of primes in the sequence $a+nd$ , we must eventually hit a prime larger than the prime $a$ , which starts a new sequence $a_{1}+n(d+1)$ , which repeats infinitely. Iffy★Chat -- 08:25, 30 April 2020 (UTC)[reply]