Two (happy) notes on unit fractions

After some encouragement of Ernie Croot (yes, the man that proved the Erdös-Grahamproblem), I managed to find the following generalization of an earlier shown theorem;

Let n and m be any given natural numbers and p be a given prime. Let a \in \aleph be the unique number not divisible by p and smaller than p^m for which a k \in \aleph_0 exists such that ap^k is the closest underapproximation of n possible**. Let L_n be the least common multiple of the first n natural numbers and let X_n be such that \dfrac{X_n}{L_n} = \displaystyle \sum_{i=1}^n \dfrac{1}{i}. Then: p^m divides X_n if, and only if, p^m divides X_a.

On another happy note, on Wolfram Mathworld I found the following conjecture (slightly generalized by myself):

Let D_n be the number of digits of the denominator of the n-th harmonic number, written in base b. Then: \dfrac{n}{D_n} converges to \log{b}.

And I think the best application of everything I have been able to show so far concerning the problems of Erdös and Graham, is an easy proof of the above (sweet!) conjecture 🙂

** Since this definition might be vague, here are some examples:
If n = 100, p = 7 and m = 1, we have a = 2 
(2*7^2 = 98 \le 100). So our theorem says 7 | X_{100} iff 7 | X_2, which is not the case.
If n = 100, p = 7 and m = 3, we have a = 100
(100*7^0 = 100 \le 100). So our theorem says 7^3 | X_{100} iff 7^3 | X_{100}, which is not the case.
If n = 100, p =5 and m = 2, we have a = 4 (4*5^2 = 100 \le 100). So our theorem says 5^2 | X_{100} iff 5^2 | X_4, which is the case. In general we have (assuming n > p^m, otherwise a would just equal n and our theorem then trivializes):
k = \left\lfloor\log_p{n}\right\rfloor - m + 1,
a' = \left\lfloor\dfrac{n}{p^k}\right\rfloor and
a = \dfrac{a'}{(a', p^m)}


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