My dd3 is like this, very hard to entertain while her two sisters are at school. I think it's partly because neither of us is used to the one-adult-one-child dynamic which so many people are very used to - my first two children were very close together. So dd3 doesn't know how to entertain herself alone and I don't really bother to do stuff with her cos I'm used to her sisters entertaining her.
For us it was only one day a week as I worked the other days but my solution was to up her preschool hours and my work hours. We were both happier. She had children to play with and I wasn't being yelled at all day.
DD and I go out most mornings to get rid of her 'boredom energy' ... Parks/swings/playgroups/swimming/people's houses etc etc. Then home at lunchtime (say 12.30), then she seems pretty chilled until 3 when we set off for pick up. I try to stay home one morning a week - I feel it important she learns to be bored at home, sounds strange but she does seem to think up stuff to do after initial moaning. I don't tend to give in to the 'wanting my attention all the time' thing ... I did that with DS and he still finds it hard to entertain himself and he is nearly 9
My 3 are really close and I think she's missing them quite badly, she's regressed a bit on the toilet front(wet pants) too, just won't settle and is generally being a mare. She does 4 sessions so don't want to up it(she's my last) but last week I could happily have booked her in full time if it was possible I think the boredom factor is good but crikey she's hard work at the moment.