At the PureCN github repository, when you choose the Branch Master drop-down you see "Release_3_5", "Release_3_6", etc. which don't correspond at all to PureCN versions, which all start with "1.", so I'm wondering how to interpret those? How would I determine the PureCN version number of the latest version on the release branch in the repository (without cloning the repository, building it, loading it, and looking at the version number)?

Those 3_X versions refer to the version of bioconductor that they were release for.

For instance, the current version of bioconductor is 3.10, which you can see from the News section of the home page, or up top on the Install page.

When you navigate to the package home page for PureCN, you'll see that you can see "release" in the URL there, which corresponds to the version release of Bioconductor (3.10) [it even says so under the description of the package). This corresponds to a version 1.16.0 of PureCN (look in the Details section, there is a "Version" row).

(FYI: you can hack the URL to see what version of PureCN was using in the previous version of Bioconductor by changing "release" to 3.9, like so: http://bioconductor.org/packages/3.9/bioc/html/PureCN.html)

The fact that you have v1.17.1 installed suggests that you somehow installed the development version of this package.

If you're not sure how you got here suggests that perhaps a few things might be haywire with your installation. Maybe you can start by running BiocManager::valid() to see if it gives you any helpful output. If you're having problems parsing that, perhaps you can update your question by copy/pasting its results in the question itself.

Also, if you want to understand how versioning works in the bioconductor universe, you can read through the Version Numbering page which goes through the scheme in more detail.

When I look at the Bioc devel webpage for PureCN (http://bioconductor.org/packages/devel/bioc/html/PureCN.html) I see version 1.17.1.

The RELEASE_3_5, Release_3_6 etc branches correspond to Bioconductor releases rather than PureCN specifically. The most recent release is RELEASE_3_10. These branches indicate how things are organised on the Bioconductor git repository, and in addition to the RELEASE_3_X branches each package also has a master branch which is what constitutes the developmental version of Bioconductor.

Generally speaking the second part of a packages version number increases by two for each release of Bioconductor. So for PureCN 1.16.Z corresponds to release 3.10 and 1.14.Z to release 3.9. However there's no general rule for mapping between a package version and a Bioconductor release - it will depend upon when a package was first added to Bioconductor.