Sum of the reciprocal of the prime factors of a square free number

Question

Let $n$ an odd square free number, and $p_1, \ldots , p_n$ their distinct prime factors. Ir is true that
$$ \sum\limits_{i=1}^n \frac{1}{p_i} < 1? $$

Otherwise, there exists some conditions to ensure that this works?

Answer 1

The sum of the reciprocals of the primes diverges, see, e.g., this question, thus the answer to your question is No.

Mind you, it diverges very slowly. The first odd counterexample is $$N= 3234846615=3\times 5\times 7\times 11\times 13\times 17\times 19\times 23\times 29 $$

For that number, the sum comes to just over $1$ (see this). But don't let the slow speed of divergence fool you. If you put in enough primes you can make the sum as large as you want.