Augment (3x needed for Caldesann’s Despair Recipe in Kanai’s Cube = initial gems needed): (81x Imperials) produces (27x Flawless Imperials) produces (9x Royals) produces (3x Flawless Royals).
Single Flawless Royal (initial gems needed): 27x Imperials) produces (9x Flawless Imperials) produces (3x Royals) produces (1x Flawless Royal)
If you are running low on the gem types you use, the merchant Squirt the Peddler in Act 2 Hidden Camp sells Cube gem conversion recipes (Essence of XYZ … where XYZ is the gem type). These recipes can be used to change gems to the desired type (in groups of 9). E.g., take your stack of unused Topaz, throw it into the cube with a Essence of Ruby recipe, and it will remove 9 Topaz and give you 9 Rubies.
These recipes are not limited to Augments. They are also useful for crafting Flawless Royals for any armor or weapons socket slots.