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3

Part of 2016 Bosnia and Herzegovina Team Selection Test

Problems(1)

Multiplies of p in increasing sequence

Source: Bosnia and Herzegovina TST 2016 day 1 problem 3

5/16/2016
For an infinite sequence a1<a2<a3<...a_1<a_2<a_3<... of positive integers we say that it is nice if for every positive integer nn holds a2n=2ana_{2n}=2a_n. Prove the following statements:
a)a) If there is given a nice sequence and prime number p>a1p>a_1, there exist some term of the sequence which is divisible by pp.
b)b) For every prime number p>2p>2, there exist a nice sequence such that no terms of the sequence are divisible by pp.
number theoryprime numbersInteger sequence