This is interesting to me too. In my case, I decided for a simple approach: by posting on the arXiv, here is [one such paper][1]. I also created a separate route on my own website. Since then the conjectures have received several attention as you can see [at these coordinates][2] where I keep updating new papers proving the claims or certain partial progress. That is one possible choice though, you may still opt to find a home for your work in some journals, such as in Experimental Mathematics as Carlo Beenakker pointed out. Perhaps otheres can name the journals they like suggesting.



[1]: http://front.math.ucdavis.edu/1207.4045
[2]: http://dauns01.math.tulane.edu/~tamdeberhan/conjectures.html