We have experimentally determined the production rate of $^{39}$Ar and $^{37}$Ar from cosmic ray neutron interactions in argon at sea level. Understanding these production rates is important for argon-based dark matter experiments that plan to utilize argon extracted from deep underground because it is imperative to know what the ingrowth of $^{39}$Ar will be during the production, transport, and storage of the underground argon. These measurements also allow for the prediction of $^{39}$Ar and $^{37}$Ar concentrations in the atmosphere which can be used to determine the presence of other sources of these isotopes. Through controlled irradiation with a neutron beam that mimics the cosmic ray neutron spectrum, followed by direct counting of $^{39}$Ar and $^{37}$Ar decays with sensitive ultra-low background proportional counters, we determined that the production rate from cosmic ray neutrons at sea-level is expected to be $(759 pm 128)$ atoms/kg$_text{Ar}$/day for $^{39}$Ar, and $(51.0 pm 7.4)$ atoms/kg$_text{Ar}$/day for $^{37}$Ar. We also performed a survey of the alternate production mechanisms based on the state-of-knowledge of the associated cross-sections to obtain a total sea-level cosmic ray production rate of $(1048 pm 133)$ atoms/kg$_text{Ar}$/day for $^{39}$Ar, $(56.7 pm 7.5)$ atoms/kg$_text{Ar}$/day for $^{37}$Ar in underground argon, and $(92 pm 13)$ atoms/kg$_text{Ar}$/day for $^{37}$Ar in atmospheric argon.
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