Graham the antisocial giant had 201 grandchildren. "Too many grandchildren!" he thought to himself glumly, although he was aware that this wasn't the sort of thing you were really meant to say about your grandchildren. "Still, I suppose I'd better see them all soon, probably in the first 60 days of the year" he said.
Graham arranged to meet his first grandchild on the second day of the year, his second grandchild on the 4th day of the year, his $n$th grandchild on the $2n$th day of the year and so on up to and including day 60. That dealt with the first 30 of them. Then he arranged to meet his 31st grandchild on the 3rd day of the year, his 32nd grandchild on the 6th day of the year, his $(30+m)$th child on the $3m$th day of the year and so on. He continued this way, slotting in grandchildren on days numbered with multiples of $4$, then of $5$, all the way up to multiples of $60$. Handily that took care of them all.
"Now" said Graham to himself, "on any day I that I have an even number of grandchildren visiting me it's fine, because they can all pair off and talk to each other. But if I have an odd number of grandchildren visiting then I'll have to be sociable. This seems as good a time as any to learn about Euler's totient function".
On how many of the 60 days will Graham have to be sociable?