# Solution to puzzle 34: Harmonic sum 2

Let H_{n} = 1/1 + 1/2 + ... + 1/n.

Show that, for n > 1, H_{n} is not an integer.

Consider, for n > 1, H_{n} = 1/1 + 1/2 + ... + 1/n, expressed as a fraction, a/b, where b is the least common multiple of 2, 3, ..., n.

Then b = 2^{r}· s, where 2^{r} is the largest power of 2 less than or equal to n, and s is an odd number.

As an example, consider a and b, for H_{5}:

b = 2^{2}· s. (In fact, s = 15.)

a = 2^{2}· s + 2s + 2^{2}· (s/3) + s + 2^{2}· (s/5).

Hence a is odd, as it is the sum of one odd number, s, and several even numbers.

Clearly this argument can be generalized.

For any H_{n}, n > 1, a is the sum of a single odd number, coming from the largest power of 2 less than or equal to n, and n − 1 even numbers.

Therefore a is odd.

Obviously, b is even.

Hence none of the twos in the prime factorization of the denominator can be cancelled from the numerator.

So, when a/b is written as a fraction in its lowest terms, c/d, d > 1.

Therefore, for n > 1, H_{n} is not an integer.

## Generalization

Similarly, 1/m + ... + 1/n is never an integer, for 1 < m < n.

## Further reading

- Online Encyclopedia of Integer Sequences: A025529, A001008, A002387.

Source: Mauro Maggioni. (Puzzle page since taken down.)

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