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Hometable of small multiply perfect numbers

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# table of small multiply perfect numbers

The first five $k$-multiperfect numbers for $1<k<6$ 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 | 6 | 28 | 496 | 8128 | 33550336 |
---|---|---|---|---|---|

3 | 120 | 672 | 523776 | 459818240 | $11(2^{{27}}-2^{{13}})$ |

4 | 30240 | 32760 | 2178540 | 23569920 | 45532800 |

5 | 14182439040 | 31998395520 | 518666803200 | 13661860101120 | 30823866178560 |

6 | $297581328(5\textrm{-}P_{3})$ | $\displaystyle\left(\frac{1845}{31}\right)6\textrm{-}P_{1}$ | $\displaystyle\left(\frac{27335}{369}\right)6\textrm{-}P_{2}$ | $\displaystyle\frac{(13\#)210^{{18}}412057}{3^{9}5^{1}67^{1}718241}$ | $\displaystyle\frac{(13\#)2^{{18}}105^{5}793}{58339155}$ |

The smallest 7-multiperfect number is 14131089794743834825984940273848552326434354481 8565120000.

The source for $1<k<6$ 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.

## Mathematics Subject Classification

11A05*no label found*

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