(abridged) [...] Methods: In a continued study of the molecular core population of the Pipe Nebula, we present a molecular-line survey of 52 cores. Previous research has shown a variety of different chemical evolutionary stages among the cores. Using the Mopra radio telescope, we observed the ground rotational transitions of HCO+, H13CO+, HCN, H13CN, HNC, and N2H+. These data are complemented with near-infrared extinction maps to constrain the column densities, effective dust temperatures derived from Herschel data, and NH3-based gas kinetic temperatures. Results: The target cores are located across the nebula, span visual extinctions between 5 and 67 mag, and effective dust temperatures (averaged along the lines of sight) between 13 and 19 K. The extinction-normalized integrated line intensities, a proxy for the abundance in constant excitation conditions of optically thin lines, vary within an order of magnitude for a given molecule. The effective dust temperatures and gas kinetic temperatures are correlated, but the effective dust temperatures are consistently higher than the gas kinetic temperatures. Combining the molecular line and temperature data, we find that N2H+ is only detected toward the coldest and densest cores while other lines show no correlation with these core properties. Conclusions: Within this large sample, N2H+ is the only species to exclusively trace the coldest and densest cores, in agreement with chemical considerations. In contrast, the common high-density tracers HCN and HNC are present in a majority of cores, demonstrating the utility of these molecules to characterize cores over a large range of extinctions. The correlation between the effective dust temperatures and the gas kinetic temperatures suggests that the former are dominated by dust that is both dense and thermodynamically coupled to the dense gas traced by NH3. [...]
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