4444 + 4446 + 4448 + 4454 + 4456 + 4458 + ............. + 9984 + 9986 + 9988 + 9994 + 9996 + 9998 = 4,679,208

I guess an expert at Number Theory could come up with some exotic equation;

all I can see (if it's any help) is the repeating 4-6-8 pattern: 444(4) 444(6) 444(8)....