In layman terms, Old Atlus was a division of Index Corporation, but after Index Corporation became bankrupt, their remaining Atlus and Index assets were sold to Sega and absorbed into a newly formed subsidiary called Sega Dream Corporation.
Sega Dream Corporation was later rebranded as Index Corporation and then rebranded again to Atlus with all the non-gaming assets of Sega Dream Corporation being spun-off into a new company called Index Corporation.
Sega has announced, via a press release, that they will split their subsidiary, Index, into Atlus and Index on April 1.
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Index Corporation was eventually sold off to Sawada Holdings because the company was no longer relevant within Sega Sammy's conglomerate.