Classes of two-nucleon ($2N$) contact interactions are developed in configuration space at leading order (LO), next-to-leading order (NLO), and next-to-next-to-next-to-leading order (N3LO) by fitting the experimental singlet $np$ scattering length and deuteron binding energy at LO, and $np$ and $pp$ scattering data in the laboratory-energy range 0--15 MeV at NLO and 0--25 MeV at N3LO. These interactions are regularized by including two Gaussian cutoffs, one for $T,$=$,0$ and the other for $T,$=$,1$ channels. The cutoffs are taken to vary in the ranges $R_0,$=$(1.5$--2.3) fm and $R_1,$=$(1.5$--3.0) fm. The 780 (1,100) data points up to 15 (25) MeV energy, primarily differential cross sections, are fitted by the NLO (N3LO) models with a $chi^2$/datum about 1.7 or less (well below 1.5), when harder cutoff values are adopted. As a first application, we report results for the binding energies of nuclei with mass numbers $A,$=$,3$--6 and 16 obtained with selected LO and NLO $2N$ models both by themselves as well as in combination with a LO three-nucleon ($3N$) contact interaction. The latter is characterized by a single low-energy constant that is fixed to reproduce the experimental $^3$H binding energy. The inclusion of the $3N$ interaction largely removes the sensitivity to cutoff variations in the few-nucleon systems and leads to predictions for the $^3$He and $^4$He binding energies that cluster around 7.8 MeV and 30 MeV, respectively. However, in $^{16}$O this cutoff sensitivity remains rather strong. Finally, predictions at LO only are also reported for medium-mass nuclei with $A,$=$,40$, 48, and 90.
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