What is the significance of the the thirteenth, twenty-first, twenty-second, and twenty-seventh days of the month of Rajab?

There is nothing specific to certain days in Rajab with particular acts of worship. The month of Rajab is revered because it is one of the Sacred Months. The thirteenth day of every month, not just Rajab, is one of the Ayyam al-Bid (White Days) on which fasting is recommended. Designating specific days for worship without legitimate religious evidence is considered an innovation (bid'ah).