In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order k, expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.

Linear complexity profile and correlation measure of order k are important pseudorandomness measures for sequences used in cryptography. We study both measures for a class of binary sequences called Legendre-Sidelnikov sequences. The proofs involve character sums.