Natural numbers to exponents

I wanted to find a way to relate the natural numbers to the exponents. Below you can see this relationship.

What I did:
I added the natural sequence (1,2,3,4….) below the (blue) exponential sequence. With each “shortcoming” like at 4 (3+…) I added a new sequence of natural numbers below. Until the sum of these natural number sequences becomes the same value as the exponential sequence.

The funny thing is, the addition of natural sequences is also exponential but with a delay of two, for example, at 4 you add 1 natural sequence, at 8 you add 2, at 16 you add 4, at 32 you add 8 and so on. I only did this until 128. 128 is correct too, but not completely on the picture (due to picture resolution stuff).

I’m not sure whether there is any practical implication of this “thing” I’ve made, but it seemed quite a funny coincidental finding to post it on my website.

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