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Disproof of a conjecture of Jacobsthal

For any integer n ≥ 1, let j(n) denote the Jacobsthal function, and ω(n) the number of distinct prime divisors of n. In 1962 Jacobsthal conjectured that for any integer r ≥ 1, the maximal value of j(n) when n varies over N with ω(n) = r is attained when n is the product of the first r primes. We show that this is true for r ≤ 23 and fails at r = 24, thus disproving Jacobsthal’s conjecture.