Self-referential sequence that is sometimes powers of two

Question

I've created an integer sequence where, after the first two elements, every element is calculated using the previous two. If the first two numbers are $1$ and $3$, the sequence goes as follows:

$$1, 3, 6, 8, 12, 15, 20, 22, 33, 36, 38, 57, 60, 62, 93, 96, 98, 105, 108$$

If the sequence starts with $4$ and $6$, however, the sequence is instead this:

$$4, 6, 9, 18, 20, 24, 27, 36, 38, 57, 60, 62, 93, 96, 98, 105, 108, 110, 115$$

If the sequence starts with $1$ and $2$, it goes:

$$1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144$$

The sequence is infinite, but there are certain starting numbers that are invalid, including when the first two numbers are equal. The sequence does not rely on the base and individual digits of the numbers, nor real-world things like seven-segment displays or letters.

I have three questions:

• What comes after $115$ in the second sequence, and why?
• If the sequence goes $2, \_\_, \_\_, 16, 20$, which numbers should fill in the blanks?
• What characterizes the first two elements of sequences generating powers of two?

Answer 1

The sequence is defined as:

Let $a,b$ be the two previous numbers. The next number is $b$ plus the smallest divisor of $b$ that isn't a divisor of $a$.

For example,

In the second sequence, say so far we have $4,6,9,18,20$. The last two numbers are $a=18$, $b=20$. The divisors of $b$ are $1,2,4,5,10,20$, of which $4,5,10,20$ are not divisors of $a$. We take the smallest one, $4$, and add it to $b$ to get the next number $24$.

This explains the note that we the sequence can't start with two of the same number because

the divisors of the two previous numbers are the same, so there's no divisor to add to continue the sequence. This would also happen is the second number were a divisor of the first.

To answer the questions:

• What comes after $115$ in the second sequence, and why?

The previous two elements are $a=110$ and $b=115 = 5 \cdot 23$. Since $a$ is divisible by $5$, we take $23$ and the next number is $115 + 23 = 138$.

• If the sequence goes $2, \_\_, \_\_, 16, 20$, which numbers should fill in the blanks?

It's $7$ making $2, 7, 14, 16, 20$. I tried some values until one worked.

• What characterizes the first two elements of sequences generating powers of two?

Any two consecutive powers of $2$. This causes the only divisor of $b$ but not $a$ to be $b$ itself, so the next value is $b+b$ continuing the powers of two. Two non-consecutive powers of $2$ won't do this.