Estimated federal tax payment schedule follows interesting 2-3-4 months pattern after 1st quarter

I had to pay some estimated taxes for realized long term capital gains I had during last quarter of year. I was under impression that we typically need to pay at end of each quarter with 15 days grace period after quarter ends. But, it's not necessarily true. It seems to be following interesting 2,3,4 months schedule as below:

Source: https://www.irs.gov/faqs/estimated-tax/individuals/individuals-2

First quarter of year does follow end of quarter plus 15 days rule. Later due dates don't follow calendar quarterly schedule at all. They follow 2 months, 3 months and then 4 months period segments.

We can pay federal estimated taxes by going to IRS tax account login (using ID.me login now) here.

My state MA estimated tax payment follows same schedule as above one for federal. Perhaps other states do same. 

We can pay MA state estimated tax by going to MassTaxConnect - MA DOR Estimated Tax Payment site here. MA tax rate is 5% of taxable income.

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