We use new X-ray data obtained with the Nuclear Spectroscopic Telescope Array (NuSTAR), near-infrared (NIR) fluxes, and mid-infrared (MIR) spectra of a sample of 24 unobscured type 1 active galactic nuclei (AGN) to study the correlation between various hard X-ray bands between 3 and 80 keV and the infrared (IR) emission. The IR to X-ray correlation spectrum (IRXCS) shows a maximum at ~15-20 micron, coincident with the peak of the AGN contribution to the MIR spectra of the majority of the sample. There is also a NIR correlation peak at ~2 micron, which we associate with the NIR bump observed in some type 1 AGN at ~1-5 micron and is likely produced by nuclear hot dust emission. The IRXCS shows practically the same behaviour in all the X-ray bands considered, indicating a common origin for all of them. We finally evaluated correlations between the X-ray luminosities and various MIR emission lines. All the lines show a good correlation with the hard X-rays (r>0.7), but we do not find the expected correlation between their ionization potentials and the strength of the IRXCS.