Claims

../

2016_07_11_shiu: Shiu's 2016 preprint proves that the reduced denominator of the n-th harmonic number is smaller than lcm(1, ..., n) for infinitely many n, with the exact prime-by-prime criterion; the second question is answered yes.

2022_01_26_wu_yan: Wu and Yan's 2022 paper proves, assuming that the numbers 1/log q over distinct primes q are linearly independent over Q, that (a_n, L_n) > 1 on a set of upper density 1; the hypothesis is open, so the page derives nothing.

2024_11_07_steinerberger: Steinerberger's observation, recorded in the site's commentary, that 3 divides (a_n, L_n) whenever the leading digit of n in base 3 is 2; it answers the second question of Problem 291, that (a_n, L_n) > 1 infinitely often, yes.

2026_02_05_van_doorn: Van Doorn's note of February 2026 proves that for every periodic sequence of non-zero numerators the numerator over the lcm shares a factor with it infinitely often, with a Lean proof by Aristotle; it covers the second question.

---