Microsoft has published at December 2016 a new version of the Sync-client OneDrive for Business, most people knows his common name: Groove.exe. You still need the Sync-Client to sync data from SharePoint on Premise.

The new Sync Client (Next Generation Snyc Client == NGSC) is still in Preview, and with the latest version you are only may sync with document libraries from Office 365. Microsoft has announced  the global availability (GA) to the end of the year. Here you may look for the Next Generation Sync Client

KB3127977 Office 2013 KB3127993 Office 2016

There is only a small fix, because the Groove.exe may crash during startup

Both Updates are valid only for MSI Installations, not Click to run.

Before 12/1/2016	After 12/1/2016

Be aware that the update in Download Center applies to Microsoft Installer (.msi)-based edition of Office 2013. It doesn’t apply to the Office 2013 Click-to-Run editions such as Microsoft Office 365 Home

To determine whether your Office installation is Click-to-Run or MSI-based, follow these steps:

1. Open an Office 2013 application, such as Microsoft Word 2013.
2. On the File menu, select Account.
3. For Office 2013 Click-to-Run installations, an Update Options item is displayed. For MSI-based installations, the Update Options item isn’t displayed.

You will found the file groove.exe here:  X:\Program Files\Microsoft Office\Office15 for Office 2013 and  X:\Program Files\Microsoft Office\Office16 for Office 2016

So the version for Office 2013 of the file should be 15.0.48831000 or higher.
And the version for Office 2016 of the file should be 16.0.4471.1000 or higher.