Testing for admissible sequences
Brachylog, 25 24 19 bytes
5 bytes thanks to Karl Napf.
lybb'(eM-yA,?:[M]z:%aodA)l:2'(eM-yA,?:[M]z:%aodA)l:2'(eMg:?rz:%adlM)
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l:2'(eMg:?rz:%adlM)
l:2 Temp = [2:length(input)]
'( ) true if the following cannot be proven:
eM M is an element of the interval
indicated by Temp, i.e. from 2
to the length of input inclusive,
g:?rz:%adlM every element of input modulo M
de-duplicated has length M.
Jelly, 10 bytes
JḊðḶḟ%@ð€Ạ
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Input: L.
JḊ Range of [2..len(L)].
ð ð€ For x in [2..len(L)]:
Ḷ [0..x-1] (residue classes)
ḟ without elements from
%@ L % x.
Ạ All truthy (non-empty)?
Python, 61 60 bytes
q=lambda l,d=2:d>len(l)or q(l,d+1)&(len({v%d for v in l})<d)
All test cases on ideone
Edit: replaced logical and with bitwise & to save one byte