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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.

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    $\begingroup$ Sum of prime factors. $\endgroup$ – barak manos Apr 5 '16 at 5:04
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    $\begingroup$ @barakmanos thank you and my apologies, while you wrote the comment I added to clarify that should be strictly increasing, as all the samples I wrote. $\endgroup$ – iadvd Apr 5 '16 at 5:09
  • $\begingroup$ How exactly are the gaps between prime numbers "strictly increasing"??? $\endgroup$ – barak manos Apr 5 '16 at 5:10
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    $\begingroup$ @barakmanos strictly increasing integer sequences whose gaps between the consecutive elements are pseudorandom... the "strictly increasing integer sequence" is the prime number sequence itself... 2,3,5,7, etc. as the rest of sequences of the list. $\endgroup$ – iadvd Apr 5 '16 at 5:14
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    $\begingroup$ In that case, perfect numbers. $\endgroup$ – barak manos Apr 5 '16 at 5:18
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How about numbers that are the product of two distinct primes. 6, 10, 14, 15, 21, 22, ...

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    $\begingroup$ From OP's comment on the question itself (#6 in the comment-thread): "looking for other sequences not related with the ones above or combinations of them as I wrote in the last paragraph". $\endgroup$ – barak manos Apr 6 '16 at 6:49
  • $\begingroup$ @paw88789 thank you! as Barak said I am looking for another sequences not related with the ones I listed as stated in the question. $\endgroup$ – iadvd Apr 6 '16 at 9:35
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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 $b$.

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    $\begingroup$ I think OP would like things where the gap size does not blow up so much. Maybe take your idea mod some number of appropriate size, and use the fact that the multiples of pi is dense in the circle or something. $\endgroup$ – Alfred Yerger Apr 6 '16 at 6:40
  • $\begingroup$ @AlfredYerger: Gaps between prime numbers also eventually "blow up" (well, you should really define "blow up" mathematically, but in general, there is no limit on the gap between two consecutive prime numbers). BTW, my alternative suggestion at the bottom of the answer is not correct. For example, consider the irrational number $0.102001000200001000002000000\dots$. So I will remove it shortly. $\endgroup$ – barak manos Apr 6 '16 at 6:44
  • $\begingroup$ I don't think I was talking about prime number gaps. I just meant take your sequence mod some number of whatever size OP wants the values of the sequence to be in. Then maybe you can show that since the values of the multiples of pi mod 1 are dense in the interval, you can show that the values of $10^k \pi$ mod $c$ are dense in $[0,c]$. Then this should also satisfy OP's request. $\endgroup$ – Alfred Yerger Apr 6 '16 at 6:46
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    $\begingroup$ @AlfredYerger: You weren't, but OP has mentioned it as an example of what he/she are looking for. $\endgroup$ – barak manos Apr 6 '16 at 6:47
  • $\begingroup$ My suggestion is inappropriate anyway. The OP wants strictly increasing sequences. $\endgroup$ – Alfred Yerger Apr 6 '16 at 6:49
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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.

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