Can the difference between consecutive even abundant numbers exceed 6?
Can the difference between consecutive even abundant numbers exceed 6?
I came across an astonishing observation :
An abundant number is a positive integer $n$ with the property $S(n)>n$ , where $S(n)$ is the sum of the divisors of $n$ except $n$ itself.
The difference of consecutive even abundant numbers seems to be at most $6$. Can anyone prove/disprove this statement ?
Difference $6$ is not exceeded upto at least $10^7$.
1 Answer
1
This can be proven, observing that any multiple of $6$ greater than $6$ itself is abundant: the divisors include at least $1, fracn2,fracn3,fracn6$, which together sum to $n+1$.
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Well done! (+1)
– Peter
Aug 26 at 18:47