Consider the following sequence of numbers

What number comes next? The OEIS has no sequence beginning with these, which seems puzzling enough, but wait! There’s more.

You might, like me, wonder if this sequence contains all perfect numbers. Alas, is not in this list. Does it contain all triangular numbers?

Update: this list still confuses me. It definitely contains all squares and pronics, but it also appears to contain all numbers of the form . Also, a big fat “no” concerning the triangular numbers: is not “hydropronic.” (btw, I really need a better word for these numbers.)

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One observation: this sequence appears to be closed under multiplication. Someone should write some code that checks this. Also, the “Sprague” formula is 100% correct.

I retract my statement that this is closed under multiplication. 3 and 9 are in the list, but 27 is not. Boo.