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We show that the sequence $(\alpha n)_{n\in \mathcal{B}}$ is uniformlydistributed modulo 1, for every irrational $\alpha$, provided $\mathcal{B}$belongs to a certain family of integer sequences, which includes the prime,almost prime, squarefree, practical, densely divisible and lexicographicalnumbers. We also give an estimate for the discrepancy if $\alpha$ has finiteirrationality measure.

Uniform distribution of $\alpha n$ modulo one for a family of integer sequences