I created some feature layers from Analysis -> Use Proximity -> Find Nearest. I added the resultant feature layers to a web map and then tried using that web map in Collector.
I enabled sync on these feature layers so that my web map could be used in offline mode. They were also shared with the appropriate groups.
When I go to download the web map in Collector (for Android, version 18.0.3, build 1033), it prompts me to enter credentials for these feature layers. I tried entering credentials, but they didn't work and the map failed to download.
My work around was to create hosted feature layer views of the feature layers that were created from the Analysis tool. When I added the views to the web map, I was no longer prompted to enter credentials.
Any ideas why these layers created from Analysis prompt for credentials?