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Just a conjecture, but here it is: For any odd integer P if R=(2^(P/2)=P-1 then P is a unitary prime meaning its sequence length is P-1. If R equals 1 then P is a non-unitary prime with an unknown number of principle segments. The division indicated is an integer division where the remainder is thrown away.
This is very simialar to the standard prime test, but using the standart prime test you have to factor P-1 to find out if P is unitary.
Here's a link to the code I used to test this conjecture for odd integers from 3 to 10,000: