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|a Results are presented of searches for continuous gravitational waves from 20 accreting millisecond X-ray pulsars with accurately measured spin frequencies and orbital parameters, using data from the third observing run of the Advanced LIGO and Advanced Virgo detectors. The search algorithm uses a hidden Markov model, where the transition probabilities allow the frequency to wander according to an unbiased random walk, while the $\mathcal{J}$-statistic maximum-likelihood matched filter tracks the binary orbital phase. Three narrow sub-bands are searched for each target, centered on harmonics of the measured spin frequency. The search yields 16 candidates, consistent with a false alarm probability of 30% per sub-band and target searched. These candidates, along with one candidate from an additional target-of-opportunity search done for SAX J1808.4$-$3658, which was in outburst during one month of the observing run, cannot be confidently associated with a known noise source. Additional follow-up does not provide convincing evidence that any are a true astrophysical signal. When all candidates are assumed non-astrophysical, upper limits are set on the maximum wave strain detectable at 95% confidence, $h_0^{95\%}$. The strictest constraint is $h_0^{95\%} = 4.7\times 10^{-26}$ from IGR J17062$-$6143. Constraints on the detectable wave strain from each target lead to constraints on neutron star ellipticity and $r$-mode amplitude, the strictest of which are $\epsilon^{95\%} = 3.1\times 10^{-7}$ and $\alpha^{95\%} = 1.8\times 10^{-5}$ respectively. This analysis is the most comprehensive and sensitive search of continuous gravitational waves from accreting millisecond X-ray pulsars to date.
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