The approximation $$H_n\approx log(2n+1)$$ https://math.stackexchange.com/a/1602945/134791

suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic numbers for its factors.

When checking for these distances, the following results are obtained:

$$ e^{H_1+H_3} \approx 17.002 $$ $$ 2e^{H_1+H_6} \approx 63.00078 $$ $$ 2e^{H_4+H_4} \approx 129.000186 $$

although the expected numbers are $$ e^{log\left(3\right)+log\left(7\right)} = 21 $$ $$ 2e^{log(3)+log\left(13\right)} = 78 $$ $$ 2e^{log\left(9\right)+log\left(9\right)}= 162 $$

The results are far from expected, but the relationship does not seem random. The first observation $$ e^{H_1+H_3} =e^{1+\frac{11}{6}}=e^{\frac{17}{6}}\approx 17.002 $$ is equivalent to $$ \frac{17}{log(17)}\approx6$$ which has the same form as $$\frac{163}{log(163)}\approx32$$ (equation 21 in http://mathworld.wolfram.com/AlmostInteger.html)

Similarly, $$\frac{90}{log(90)}\approx20$$ which is equivalent to $$e^{3H_2}\approx90$$

The third almost-integer appears in an expression very similar to $e^\pi-\pi$: $$2+\frac{e^{2H_4}}{2\left(log(2)+log(3)\right)}\approx 19.99909$$

Q: How can these almost-integers be explained?


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