I want to generate N, say N=10, random numbers, then turn off Stata, start a new instance of Stata, generate another set of N random numbers, and so on. I repeat the process R times, say R=2.

I want my R sets of N random numbers to be independent/uncorrelated. I want them to be as unrelated to each other as a pseudo random number sequences can be.

How should I be setting the seed in each of my R sequences to achieve this task?

I understand that if I set the seed to the same number, say 1, at the start of each of the R sequences, I will get R replicas of the same set of N pseudo random numbers.

What I do not understand is what happens if I set the seed to 1, generate 10 random numbers, then set the seed to 5, and generate second set of 10 random numbers. Would 5 of the random numbers in the two sequences overlap? It does not appear to be the case:

Code:
. clear

. set obs 13
number of observations (_N) was 0, now 13

. set seed 10

. gen u = runiform()

. list

     +----------+
     |        u |
     |----------|
  1. | .6012831 |
  2. | .9137043 |
  3. | .2667319 |
  4. | .6073658 |
  5. | .0360737 |
     |----------|
  6. |   .75318 |
  7. | .4591727 |
  8. | .9056994 |
  9. | .3134366 |
 10. | .3755541 |
     |----------|
 11. | .1387824 |
 12. | .9243277 |
 13. | .3676381 |
     +----------+

. clear

. set obs 3
number of observations (_N) was 0, now 3

. set seed 20

. gen u = runiform()

. list

     +----------+
     |        u |
     |----------|
  1. | .7156579 |
  2. | .9087018 |
  3. | .7972254 |
     +----------+
So I can see that the 3 numbers that should overlap are not the same. But are they independent/uncorrelated as much as pseudo random number can be?