In this paper, we report the measurement relative to the production of forward neutrons in proton-proton collisions at $mathrm{sqrt{s} = 13~TeV}$ obtained using the LHCf Arm2 detector at the Large Hadron Collider. The results for the inclusive differential production cross section are presented as a function of energy in three different pseudorapidity regions: $eta > 10.76$, $8.99 < eta < 9.22$ and $8.81 < eta < 8.99$. The analysis was performed using a data set acquired in June 2015 that corresponds to an integrated luminosity of $mathrm{0.194~nb^{-1}}$. The measurements were compared with the predictions of several hadronic interaction models used to simulate air showers generated by Ultra High Energy Cosmic Rays. None of these generators showed good agreement with the data for all pseudorapidity intervals. For $eta > 10.76$, no model is able to reproduce the observed peak structure at around $mathrm{5~TeV}$ and all models underestimate the total production cross section: among them, QGSJET II-04 shows the smallest deficit with respect to data for the whole energy range. For $8.99 < eta < 9.22$ and $8.81 < eta < 8.99$, the models having the best overall agreement with data are SIBYLL 2.3 and EPOS-LHC, respectively: in particular, in both regions SIBYLL 2.3 is able to reproduce the observed peak structure at around $mathrm{1.5-2.5~TeV}$.
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