# Number $N>6$, such that $N-1$ and $N+1$ are primes and $N$ divides the sum of its divisors

The perfect number $6$ is in the middle of the primes $5$ and $7$. It is the only perfect number with this property because odd numbers are not in the middle of two twin primes and even perfect numbers have the form $2^{n-1}(2^n-1)\ ,\ n\ge 2$ with a prime $2^n-1$, so they are greater than $4$, but not divisible by $3$ for $n>2$, hence not the middle of two twin-primes.

The least multi-perfect number $N$, for which $N-1$ is prime (except $6$), is $$N=13794\ 54720=2^8\times3\times5\times7\times19\times37\times73$$

, which is $4$-perfect, that means $\frac{\sigma(n)}{n}=4$ , where $\sigma(n)$ is the sum of the divisors of $n$.

• Is there a multi-perfect-number $N$ greater than $6$, such that $N-1$ and $N+1$ are both prime ?
• If yes, what is the minimal $N$ ?

Looking at the list http://oeis.org/A007691/b007691.txt of the first 1600 multiply perfect numbers, you find that the smallest number you are looking for seems to be \begin{align} n&=1928622300236318049928258133164032 \\ &=2^{33}\cdot3^4\cdot7\cdot11^3\cdot31\cdot61\cdot83\cdot331\cdot43691\cdot131071 \end{align} which is $4$-perfect as well, with $n-1$ and $n+1$ being prime.
(The next $n$ in that sequence appears to be 20736673935772776371907300845135221460949851029611211752485268322919135405936727005685049658102947046034932245258365859207952308374437205033138073857921732017237200100173505791123852481233523248079738393931546624000000000 - don't ask me for $\sigma(n)/n$ for that one, though!)
• The large number is $8$-perfect. – Peter May 7 '15 at 22:18