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The Scorpius-Centaurus (Sco-Cen) young and nearby massive star-forming region is particularly well suited for extrasolar planet searches with both direct imaging and radial velocity (RV) techniques. The RV search, however, is challenging, as the stars are faster rotators on average than their older stellar counterparts of similar spectral types. Moreover, the RV time series show strong signatures of stellar variability (spots and faculae) and/or stellar pulsations. Our aim is to search for giant planets (GPs) and brown dwarfs at short orbital distances around star members of the Sco-Cen association. We also aim at using these data together with others available on young stars to estimate the GP occurrence rate for young stars for periods of up to 1000 days. We used the HARPS spectrograph on the 3.6m telescope at the La Silla Observatory to monitor 88 A-F Sco-Cen stars. To improve our statistics and analysis, we combined this survey with two previous surveys that focused on young nearby stars (YNS) to compute companion occurrence rates from a sample of 176 young A-M stars. We report the discovery of a massive hot-Jupiter candidate around HD 145467, together with the discovery of one probable short-period (P < 10 days) brown dwarf around HD 149790. In addition, we confirm the binary nature of eight single-line binaries: HD 108857, HD 108904, HD 111102, HD 114319, HD 121176, HD 126488, HD 126838, and HD 133574. From our sample, we obtain a GP ($m_c\in[1;13] M_{Jup}$) occurrence rate of $0.7_{-0.2}^{+1.6} \ \%$ for periods between 1 and 1000 days and a brown dwarf ($m_c\in[13;80] M_{Jup}$) occurrence rate of $0.6_{-0.2}^{+1.4} \ \%$, in the same period range. In addition, we report a possible lack of close ($P\in[1;1000] days$) GPs around young F-K stars compared to their older counterparts, with a confidence level of 95%.