A sole proprietor generally pays state income tax to state(s) where they live and/or work.
Georgia
From my understanding of your question, you are a Georgia resident. Therefore, you are liable for Georgia income tax, regardless of where the money was earned. Georgia taxes residents on income earned anywhere.
You would still owe GA income tax, for example, if you temporarily worked in Pennsylvania for several months, but still considered yourself a GA resident.
GA estimated tax form
Pennsylvania
You said your company pays your invoice "for work at their Pennsylvania factory." If you worked in PA, you also owe PA income tax. Georgia will give you credit for tax paid to another state.
If you worked remotely in GA for a PA factory, you only owe GA income tax. PA says:
Generally, PA source income is income derived from activity or property located in PA. Such income includes:
- Income from ownership or disposition of real or personal property located in PA. For example, rental income from PA property or the sale of real estate located in PA.
- Income from work performed in PA, such as personal services or business income.
- Income from intangible property employed in a trade, profession, occupation or business carried on in the Commonwealth.
- Gambling and lottery winnings from a wager placed in PA.
PA Source Income Rules
PA estimated tax form
North Carolina
A headquarters location does not affect state income tax. For example, Walmart is headquartered in Arkansas but employees only pay tax to their home and work states, not to Arkansas (unless they live or work there). You do not owe NC income tax.