My July Diary included the following brainteaser.
There is a nonempty family of positive integers N that have the following property: the sum of the primes less than the number of primes less than or equal to N is N itself.
I don't know whether the family has a name. For my purposes here I shall call them the Zimbalist Numbers.
Take 77, for example. The number of primes less than or equal to 77 is 21. They are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73.
(Mathematicians don't include the number 1 among the primes because all the theory is more elegant and straightforward if you don't. Even including 2 is sometimes a minor nuisance; that's why a lot of theorems and problems start with, "Let p be any odd prime …")
OK: there are 21 primes less than or equal to 77. The sum of the primes less than 21 is 2+3+5+7+11+13+17+19, which is 77. It's a Zimbalist Number! So here's the two-part brainteaser. Can you find any other Zimbalist numbers? How many are there: just the one, precious few, a lot, infinitely many?
Microsoft Excel to the rescue. I cooked up a quick'n'dirty spreadsheet. Here's a partial screenshot:
Since the definition is "the sum of the primes less than [something] is N," the only numbers N that are candidates for the title "Zimbalist number" are those that are the sums of all the primes up to some point. These candidate numbers are the ones I've appropriately called N in the second column. The rest of the spreadsheet explains itself.
Looks like we have just four winners: 77, 100, 129, 160. For bigger numbers, the ratio S / N just gets bigger indefinitely (although a tad unsteadily).