Agree with everything @ojs said in his reply. I can add some unnecessary detail:
A standard baritone ukulele D string should work. It won't hurt your uke.
D'Addario classical guitar strings are sold individually, and their SINGLE SILVER WOUND 036/J31 5TH HARD TENSION just might be perfect for you.
A perfect answer would take into account
- the exact scale length
- string materials and construction
- gauges/tension of the other 3 strings
- width of the slot for that string in the nut
Even then, you may prefer a different gauge than the theoretical. Strings do not behave in a perfectly theoretical way anyway, so no calculator could really give you an exact answer. Strings are often a matter of personal preference, and you may need to do some experimentation. However, you can't go wrong if you keep in mind a few of basic 'rules':
- The pitch that you want (C) is only a whole-step below the standard baritone D string, so you could get by with a standard baritone string, and the only drawback might be that it feels a little floppier than the other strings.
You probably want all strings to have the same tension (to feel equally 'tight') for playability. Note that tension and string density can also have a slight effect on correct intonation up the neck.
- To match the tension as near as possible, you would probably want a string with slightly more mass than the standard baritone string. (The wound Low D strings tend to be somewhere between '.030' to '036'.) You could achieve this by selecting from a 'hard tension' string set, or by selecting from a set with a slightly larger gauge (for example, try a '.036' rather than a '.032'). A classical guitar 'A' string is may be a great solution; they are similar or identical in composition to a baritone uke wound D string, and tend to be around '.035'/'.036' gauge. https://www.daddario.com/products/guitar/single-strings/classical-trebles/classics-trebles/item/daddario-j3105-rectified-classical-guitar-single-string-hard-tension-fifth-string-1077/
- Your main criteria should be to get approximately the same tension as the rest of your strings. As you're bringing it up to pitch, check that the strings all 'feel' the same. If you get a string with way too much mass, and try to bring it up to pitch by using super-high tension, you could damage your instrument. However, damage would probably require a string so fat that it wouldn't fit into the nut slot anyway. To be safe, just check to make sure that:
- Each string feels approximately equally tight.
- You can easily pinch two strings together until they touch.
If you're not sure of the age or condition of your other strings, you should probably go ahead and replace the whole set. Here is one mail-order option that might work well for you: https://theukulelesite.com/ukelogic-hard-baritone.html