# Two sequences in one

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?

• your sequence is know in OEIS (A004397) as well, and gives answers to both of your questions. Apr 23 '21 at 17:35
• @ThomasL can you point to the reference for part (b) as I can't see it on OEIS? Apr 23 '21 at 18:29
• @hexomino, you are right, A004397 does not answer part b), I misunderstood the question. Apr 23 '21 at 19:00
• 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)? Apr 23 '21 at 23:10
• @SheikYerbouti Yes, but not enough to show that infinitely many of these are prime, or that infinitely many are Fibonacci. Apr 23 '21 at 23:11

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.

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), ...