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Measurements of the production cross sections of prompt D0, D+, D∗+, D+s, Λ+c, and Ξ+c charm hadrons at midrapidity in proton−proton collisions at s√=13 TeV with the ALICE detector are presented. The D-meson cross sections as a function of transverse momentum (pT) are provided with improved precision and granularity. The ratios of pT-differential meson production cross sections based on this publication and on measurements at different rapidity and collision energy provide a constraint on gluon parton distribution functions at low values of Bjorken-x (10−5−10−4). The measurements of Λ+c (Ξ+c) baryon production extend the measured pT intervals down to pT=0(3)~GeV/c. These measurements are used to determine the charm-quark fragmentation fractions and the cc¯¯ production cross section at midrapidity (|y|<0.5) based on the sum of the cross sections of the weakly-decaying ground-state charm hadrons D0, D+, D+s, Λ+c, Ξ0c and, for the first time, Ξ+c, and of the strongly-decaying J/psi mesons. The first measurements of Ξ+c and Σ0,++c fragmentation fractions at midrapidity are also reported. A significantly larger fraction of charm quarks hadronising to baryons is found compared to e+e− and ep collisions. The cc¯¯ production cross section at midrapidity is found to be at the upper bound of state-of-the-art perturbative QCD calculations.