Wouter, et al:
the denominator of the sum of the first n Harmonic numbers, equals LCM(1,2,..,n) for values of n of the form n=(p-1) , p prime.
Not all primes qualify for the converse statement.
Wouter then asks, why do the qualifying and non-qualifying primes come in runs. --- Let's call the N'th harmonic number H(n), and its reduced denominator D(n). Sometimes, D(n) is less than D(n-1). It happens when N is a multiple of some prime P, and the addition of 1/n makes the numerator a multiple of P: The P's cancel. Generally, the next multiple of P restores P to the denominator. Between those multiples of P, D(n) cannot equal LCM(1..n). As an example, let's start with D(31). LCM(1..31) = D(31) = 72201776446800. H(32) introduces a new power of two in both cases, and LCM(1..32) = D(32). H(33) makes the numerator a multiple of 11, and so the factor 11 vanishes in D(33). It is not restored until H(44). In particular, D(37), D(41), and D(43) have no factor 11, and are unequal to LCM(1..37), LCM(1..41), and LCM(1..43). Meanwhile, the factor 7 was lost at H(42), and D(47) != LCM(1..47). But then H(49) restores 7, and D(53) = LCM(1..53). -- Don Reble djr@nk.ca