There are a few instances of this, starting with

1 + 2 = 3

The next occurrence is

Sum (1 to 14) = Sum (15 to 20) = 105

the remaining ones I have found are:

Sum (1 to 84) = Sum (85 to 119) = 28,561

Sum (1 to 492) = Sum (493 to 696) = 121,278

Sum (1 to 2,870) = Sum (2,871 to 4,059) = 4,119,885

Sum (1 to 16,730) = Sum (16,731 to 23,660) = 139,954,815

Sum (1 to 97,512) = Sum (97,513 to 137,903) = 4,754,343,828

Sum (1 to 568,344) = Sum (568,345 to 803,760) = 161,507,735,340

Sum (1 to 3,312,554) = Sum (3,312,555 to 4,684,659) = 5,486,508,657,735

Sum (1 to 19,306,982) = Sum (19,306,983 to 27,304,196) = 186,379,786,627,653

Sum (1 to 112,529,340) = Sum (112,529,341 to 159,140,519) = 6,331,426,236,682,470

One mathematical pattern is that the ‘split’ occurs at 1/sqrt(2)

(e.g. 159,140,519 * 1/sqrt(2) = 112,529,340.15)

To predict the next occurrence, multiply the final number of the *previous* number line by 5.828…

Well, I know this must have been done by someone before, but I was just curious if anyone knows who it was (Descartes, D’Allembert, Isaac Newton, an SDMB member?)

Does anyone know any history of this?