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The following sequence of numbers is related to two well known sequences in the OEIS, the On-Line Encyclopedia of Integer Sequences:

3, 4, 7, 10, 16, 21, 30, 40, 57, 84, 120, 181, 274, 420,...

a) Which two sequences, and how is it related to them?

b) Does the sequence itself contain infinitely many terms belonging to each of the two sequences which give rise to it?

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    $\begingroup$ your sequence is know in OEIS (A004397) as well, and gives answers to both of your questions. $\endgroup$
    – ThomasL
    Apr 23, 2021 at 17:35
  • $\begingroup$ @ThomasL can you point to the reference for part (b) as I can't see it on OEIS? $\endgroup$
    – hexomino
    Apr 23, 2021 at 18:29
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    $\begingroup$ @hexomino, you are right, A004397 does not answer part b), I misunderstood the question. $\endgroup$
    – ThomasL
    Apr 23, 2021 at 19:00
  • $\begingroup$ Primes and Fibonacci's numbers are infinite and strictly increasing. Isn't that enough of a proof that there are infinite prime(n) + Fibonacci(n)? $\endgroup$
    – anotherOne
    Apr 23, 2021 at 23:10
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    $\begingroup$ @SheikYerbouti Yes, but not enough to show that infinitely many of these are prime, or that infinitely many are Fibonacci. $\endgroup$
    – hexomino
    Apr 23, 2021 at 23:11

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Here's an argument to show that

The sequence does not contain infinitely many Fibonacci numbers

Reasoning

As $n$ gets very large the $n$th prime, $p_n$, is approximately $n \ln n$.
Meanwhile, the $n$th Fibonacci number, $F_n$, is approximately $\frac{\phi^n}{\sqrt{5}}$.
From this, it is clear that there will exist $N$ such that for all $n \geq N$, we'll have $F_{n-1} > p_n$ and hence, $$ F_n < F_n + p_n < F_n + F_{n-1} = F_{n+1} $$ hence the numbers in the sequence will always be between Fibonacci numbers after a certain point.

Not sure about the primes bit yet.

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https://oeis.org/A004397

According to OEIS, this number sequence is

a(n) = prime(n) + Fibonacci(n)

So, for example, in the first few terms, with the prime sequence being first and the Fibonacci sequence being second:

(2 + 1), (3 + 1), (5 + 2), ...

Therefore, to answer your question:

a) It's related to the Fibonacci sequence and the sequence of primes.

b) If I understand your question correctly, it does indeed contain infinitely many terms belonging to each of the two sequences, since n can go to infinity.

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  • $\begingroup$ I don't think you've proven part (b), it could be that the sequence contains finitely many primes. $\endgroup$
    – hexomino
    Apr 23, 2021 at 18:24
  • $\begingroup$ @hexomino Admittedly am not a math person, I apologize. Maybe a math person can provide more explanation here. $\endgroup$
    – Sciborg
    Apr 23, 2021 at 18:35

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