Are there arbitrarily long runs of consecutive integers whose prime factors all lie below the specified bound?

Can the product of a short run of consecutive integers contain every small prime in a prescribed range?

A primitive semiperfect number is a sum of some of its proper divisors, with no smaller divisor having that property. Does the reciprocal sum of these numbers converge?

For distinct given integers, how many distinct representatives divisible by their respective integers can be chosen in the specified interval?

Among the allowed polynomials, how small can the maximum boundary length of their sublevel sets be?

How large can the sum of reciprocal integer-logarithm weights be over a primitive set of large integers?