# table of small multiply perfect numbers

The first five $k$-multiperfect numbers for $1 are small enough to fit on a single page without the aid of horizontal scrollbars, or breaking up the numbers into more than one lines. For $k>5$, the numbers get quite big and writing them out becomes less practical. But since they tend to be divisible by lots of small primes, it makes sense to take advantage of the primorials. In the following table, the notation $n\#$ means the product of the first $n$ primes, while the notation $k\textrm{-}P_{i}$ refers to the $i$th $k$-perfect number.

2 3 4 5 6 6 28 496 8128 33550336 120 672 523776 459818240 $11(2^{27}-2^{13})$ 30240 32760 2178540 23569920 45532800 14182439040 31998395520 518666803200 13661860101120 30823866178560 $297581328(5\textrm{-}P_{3})$ $\displaystyle\left(\frac{1845}{31}\right)6\textrm{-}P_{1}$ $\displaystyle\left(\frac{27335}{369}\right)6\textrm{-}P_{2}$ $\displaystyle\frac{(13\#)210^{18}412057}{3^{9}5^{1}67^{1}718241}$ $\displaystyle\frac{(13\#)2^{18}105^{5}793}{58339155}$

The smallest 7-multiperfect number is 14131089794743834825984940273848552326434354481 8565120000.

The source for $1 in the table are the following sequences in Sloane’s OEIS: A000396, A005820, A027687, A046060 and A046061. These have all been verified with Mathematica. For larger $k$, the information comes from the Multiply Perfect Numbers Page but has not been doublechecked anew, as these numbers require far more intensive computational effort to verify.

Title table of small multiply perfect numbers TableOfSmallMultiplyPerfectNumbers 2013-03-22 17:48:21 2013-03-22 17:48:21 PrimeFan (13766) PrimeFan (13766) 16 PrimeFan (13766) Data Structure msc 11A05