Here is the full list of all 6 The Most Lecherous achievements.
You will embark on a journey to establish a harem.
SSR in hand. (Obtained SSR for the first time)
Oh my poor guy. (Draw 10 N cards in one gacha)
Brother, that's a bit too much. (Obtained at least 2 SSRs in one gacha)
You are the man among men. (Scored over 50,000 points in one mini-game)
You can really handle it. (Kept playing one mini-game for over 4 minutes)