Whether square avatars jpeg (or gif/png) reduce to 2^n or you can even 123x123?

round sizes no profit will not give
but it makes sense to make the size multiple of the size of the DCT. there seems to be 8x8 or something like that )

What's stopping you to conduct a study with measurements? Would make a good article

On the left avatar is scaled 256 -> 128. Right — 256 -> 111.
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There is a difference. But for small images, I think, be felt will not.
The main thing is to scale them proportionally. That's the proportions that the webmaster to kill a little.

there is a version that it avoids fragmentation. and full fill sectors.
Take for example the B/W image, where one pixel weighs 1 bit. Sootvetstvenno the image size of 128x128 will take 16кбит memory. While the same image size 123*123 theoretically should weigh 14.7744140625 kbps.
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emphasize the fact that the calculation of the rough-theoretical, and the reality may not have anything in common.