We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring Z_(p~n). We give conditions on the characteristic polynomial F(x) of a linear recurrence sequence u which guarantee that all elements of the ring occur in u. For the case where F(x) is a reversible Galois polynomial over Z_(p~n), we give upper bounds for occurrence indices of elements in a linear recurrence sequence u. A situation where the characteristic polynomial F(x) of a linear recurrence sequence u is a trinomial of a special form over Z_4 is considered separately. In this case we give tight upper bounds for occurrence indices of elements of u.
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