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Rumer Willis has been focused on keeping her friendships a priority ever since her mom, Demi Moore, was hospitalized back on January 23 after overdosing on cans of nitris oxide.

The 23-year-old singer ... sporting wavy red hair, basic jeans and black flats, left a friends house in LA with a notebook in hand. Perhaps she's penning some new songs?

Pretty soon, Rumer will be a big sister again. Her dad, Bruce, is expecting a baby soon with his 33-year-old wife Emma Hemming!