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Is there a standard term for the relation on sequences where two sequences are related iff they have a finite Levenshtein distance, or for the equivalence classes it induces?

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Are you referring to this levenshtein distance?. Because I think if the sequences are infinite, that levenshtein distance would imply that they both diverge or converge to the same limit. –  alejopelaez Aug 22 '11 at 21:22
    
@Peláez: Yes, that one -- are there others? –  Charles Aug 22 '11 at 21:43
    
The sequences I'm looking at will rarely have limits, so that aspect is not interesting to me. I'm more interested in seeing that, e.g., the primes are in the same class as the odd primes but in a different class than odd numbers. –  Charles Aug 22 '11 at 21:45
    
I’ve never encountered a term for it. I’d probably say that the sequences were eventually equal modulo finite shifts. –  Brian M. Scott Aug 23 '11 at 0:56
    
@Peláez: Having finite Levenshtein distance is much stronger than having the same convergence behavior: sequences $\sigma$ and $\tau$ have finite Levenshtein distance iff there are $n,m\in\omega$ such that $\sigma(n+k)=\tau(m+k)$ for all $k\in\omega$. –  Brian M. Scott Aug 23 '11 at 1:01

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