Sequences of integers might be ordered, totally ordered,... For all such attributes we find definitions in clear mathematical terms.
(1) But what is in clear mathematical terms the established and commonly accepted definition for an arbitrary sequence of integers, when it is called random (or non-random)? [Please kindly pay attention, that I am really not looking for any intuitions or thoughts or ideas (we can find lots of them on internet), rather a clear mathematical definition which I can apply in a deductive way to test an arbitrary sequence of integers to whether it is random or not.]
(2) Particularly taking into account that when we observe for the first time ever an arbitrary sequence of integers, we may not identify immediately a possible complex background of any relations or patterns behind the numbers, such relations could be however discovered at a later point of time, when sufficiently investigated. It would be paradox if such a sequence could be regarded as random for some time and non-random later. Hence my second question: Is our knowledge about a sequence a crtieria to be considered in the definition of the attribute random?
Thank you in advance!