Conjugate partitions and a duality for Josephus-type sieves, with a note on signed sums of primes

The Flavius Josephus sieve deletes every second positive integer, then every third of the remaining ones, then every fourth, and so on. We show that the same numbers survive the sieve that runs backwards over the even moduli: delete every ..., eighth, sixth, fourth entry, and finally every second one. More generally, for every c >= 1 the sieve that deletes every (ck+1)-th entry at step k = 1, 2, 3, ... and the sieve that deletes every (c+1)j-th entry for j = ..., 3, 2, 1 have the same survivors. The proof is a short argument with conjugate partitions: the numbers of entries that the two sieves delete below a fixed survivor are the row lengths and the column lengths of one Young diagram. As a consequence we prove a conjecture of M. F. Hasler (2016) relating the Flavius Josephus sieve (OEIS A000960) to the nested floor product A073359, and the related conjecture recorded in A099072. In an independent second part we prove a conjecture of K. Fujimoto (2021) on the least k such that n = ±2 ±3 ±5 ± ... ± p_k (A327467). Code for the computer verification is included.