Change the Code, Change the Sequence: Cops
Brain-Flak, 28 bytes, Distance of 4, A002817, A090809 Cracked
This answer uses 1-indexing
(({({}[()])}{}){({}[()])}{})

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For anyone interested there are 27475 valid Brain-Flak programs with Levenshtein distance 4 from this program and 27707 with distance 4 or less.  So a brute force solution would be feasible on a consumer grade computer.