From Fallen London Wiki

If your next Time the Healer with a Noman falls on January 5, to acquire a tattoo on February 9 you will have 5 Melts costing between 137 - 292 total Noman's Friend

Once you have your Noman, you'll need an additional total of 13 - 64 x Taste of Lacre to increase its Noman's Friend, which could be achieved with 5 - 22 x Unusual Pail of So-Called Snow.