Predictable random sequences depending on related seeds

I've had this idea in my head for quite a while, but I failed to articulate it in a searchable way, so I thought I'd ask it as a question. There is a lot of information about seeds and RNG algorithms, but I couldn't find much information about the relationship between the two. Many of the discussions derail into mathematical equations that seem to talk about the distribution of the series, rather then the difference between two given series and two seeds.

More to the point, I'm interested in a behaviour like this: Two different seeds producing the same 'random' sequence. In short, two different seeds (that are somewhat close to each other in the byte representation) generate very similar random number sequences. I would like to have a similar behaviour, preferably with a simple-enough condition along the lines of "the farther the two seeds are from each other the more different the two series will be".

Some additional background (in case there's a better approach that I'm completly overlooking): I'm playing around with some genetic algorithms where this sort of behaviour would be very desirble to me. I generate a few random numbers and use them to explore a very wide space of options. Later, I test those numbers out and evaluate the "goodness" of that sequence. Once I find a sequence that has a decent score, I would like to be able to generate a similar repeatable random sequence in my next generation, except I'd like it to be a little different from the first sequence, but only slightly (Thus more likely to perserve the "goodness" of the result).

I am using the seeds as the "identifier" of the sequence, so that if the first 20 numbers in the sequence are good for me, I could produce additional "good numbers" later on. Is there any algorithms where a simple relationship between the seeds will represent some sort of relationship between the series?

My apologies for the very vague question, the whole thing confuses me quite a bit...

[Disclaimer: I have an "ok" understanding of what these algorithms do , I just can't really map from a relationship between seeds to relationship on the series]