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This paper reports the results of a search for first occurrences ofsquare-free gaps using an algorithm based on the sieve of Eratosthenes. UsingQgap(L) to denote the starting number of the first gap having exactly thelength L, the following values were found since August 1999:Qgap(10)=262315467, Qgap(12)=47255689915, Qgap(13)=82462576220,Qgap(14)=1043460553364, Qgap(15)=79180770078548, Qgap(16)=3215226335143218,Qgap(17)=23742453640900972 and Qgap(18)=125781000834058568. No gaps longer than18 were found up to N=12587.

First occurrences of square-free gaps and an algorithm for their computation