# Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”

I am doing some tests with strictly increasing integer sequences whose gaps between consecutive elements show a “pseudorandom” behavior, meaning “pseudorandom” that the gaps do not grow up continuously, but they change from a bigger value to a smaller one and vice versa due to the properties of the sequence without an easy way of calculating those variations. Initially I am using the following sequences as an example:

1. Prime numbers.

2. Abundant numbers.

3. Even deficient numbers.

4. The natural numbers associated to the separated Möbius sequences $M_1$={Möbius $\mu(n)=-1$}, $M_2$={Möbius $\mu(n)=1$}, $M_3$={Möbius $\mu(n)=0$}

To continue with my tests I would require some other good examples, but I can not recall any other well known sequences with that behavior (not related with the ones above, or combinations of them).

Are there any other well known sequences in which the behavior of the consecutive gaps is “pseudorandom” in the way expressed here? Thank you!

(*) The reason of this question is the test explained here.

#### Solutions Collecting From Web of "Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”"

How about numbers that are the product of two distinct primes. 6, 10, 14, 15, 21, 22, …

Consider the sequence $a_n=\lfloor{r\cdot{b^n}}\rfloor$, where:

• $r$ is any irrational number $>1$
• $b$ is any natural number $>1$

For example, for $a_n=\lfloor\pi\cdot10^n\rfloor$ we get:

• $a_0=3$
• $a_1=31$
• $a_2=314$
• $a_3=3141$
• $a_4=31415$
• $a_5=314159$
• $a_6=3141592$
• $a_7=31415926$
• $a_8=314159265$
• $a_9=3141592653$
• $\dots$

For generally smaller gaps, use a generally small value of $b$.

Finally I found another way of obtaining this kind of sequences! Reviewing at OEIS, some kind of partition problems provide sequences whose gaps show also this kind of pseudorandom behavior I was looking for. For instance:

1. The strictly increasing elements of the multiplicative partition function: number of ways of factoring n with all factors greater than 1. Taking only the elements strictly increasing it looks like: {1,2,3,4,5,7,9,12,16,19…} and the gaps are not strictly increasing, sometimes are bigger or lower depending on the properties of the sequence.

2. Number of partitions of n into parts 5k+1 or 5k+4. Taking only the elements strictly increasing it looks like: {1,2,3,4,5,7,9,10,12,14,17,19,23…} and the gaps show the same pseudorandom behavior.