A number n is said to be refactorable, sometimes also called a tau number, if it is divisible by the number of its divisors σ_0(n), where σ_k(n) is the divisor function. The first few refactorable numbers are 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, ... (OEIS A033950). The first new n such that n and n + 1 are both refactorable numbers are 1, 8, 1520, 50624, 62000, 103040, ... (OEIS A114617). Zelinsky proved that there are no refactorable numbers a and b such that a - b = 5 and also Colton's conjecture that no three consecutive integers can all be refactorable.
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