Recall that a positive integer $n$ is callled practical if every $m=1,\ldots,n$ can be written as the sum of some distinct divisors of $n$. The only odd practical number is $1$.

In 1996 G. Melfi [J. Number Theory 56(1996), 205-210] proved that any positive even number is the sum of two practical numbers, which can be viewed as an analogy of Goldbach's conjecture. He also proved that there are infinitely many practical numbers $q$ with $q\pm2$ also practical.

In Jan. 2013 I conjectured that (cf. http://oeis.org/A209253) for each positive integer $n$ we can write $2n+1$ as the sum of a practical number and a (Sophie Germain) prime. This looks quite challenging. Let's turn to a weaker version which is an analogue of Chen's theorem for Goldbach's conjecture.

Question 1. Is it possible to prove that any sufficiently large odd number can be written as $q+P_2$ where $q$ is a practical number and $P_2$ is a prime or the product of two primes?

If this question is still difficult, then we may consider the following one.

Question 2. How to prove that any integer $n>1$ is the sum of a practical number and a positive squarefree number?

A.W. Dudek [Ramanujan J. 42(2017), 233-240] showed that any integer $n>2$ is the sum of a prime and a positive squarefree number. I conjecture that any integer $n>2$ can be written as the sum of a positive squarefree number and a practical number $q$ with $q+2$ also practical.

Your comments are welcome!


1 Answer 1


Corollary 2 of https://arxiv.org/abs/2007.11062 states that every sufficiently large odd integer is the sum of a prime and a practical number.


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