Hence 120 is a 3-fold perfect number.

Abundancy | Count | When last number was discovered | Which was last? | Are all discovered? | Estimated total number |
---|---|---|---|---|---|

1 | 1 | - | - | yes and proved | 1 |

2 | 51 | 2018-12-07 | 18.5560326 | no, there are infinitely many | ∞ |

3 | 6 | <= 1643 | 3.2049844 | yes | 6 |

4 | 36 | <= 1929 | 4.3351682 | yes | 36 |

5 | 65 | <= 1990 | 5.1744360 | yes | 65 |

6 | 245 | 1993-05-?? | 5.6720844 | yes | 245 |

7 | 516 | 1994-01-09 | 5.9403364 | almost surely yes | ~ 515 |

8 | 1136 | 2022-05-22 | 6.5244184 | probably yes | ~ 1140 |

9 | 2164 | 2022-06-05 | 7.1993123 | no | ~ 2200 |

10 | 1710 | 2022-06-11 | 7.8125224 | no | ~ 4500 |

11 | 2 | 2022-09-07 | 8.2948075 | no | ~ 10000 |

It is extremly probable, that all proper MPNs with abundancy <= 7 are discovered.

The collection of 5932 MPNs from 2022-12-31(gziped to 1121 kB) sorted by abundancy and magnitude. It is grown out of Rich's database --- thanks ---, and transformed into a new format, such that each multiply perfect number allocates one line -- an ASCII-text-record terminated by newline -- with all its additional informations in the following (standard) form:

Except of the last field,

`|`is a separator character between the fields and is not allowed inside any field.`M`indicates the abundancy of the number as a lower case letter, such that the letters a,b,c,d,... correspond to the abundancies 1,2,3,4, ... .`ln_ln`gives the decimal value of log_{e}log_{e}of the number and is rounded to 7 decimal places after the period. This serves now as a unique identifier to each number, also.`rich_id`is Rich's unique identifier for this number which encodes the abundancy appended by the exponents of at most the three primes 2,3,5. These exponents are encoded alternatingly in a 26-base system made up of the letters`a-z`und a 10-base system made up of the digits`0-9`. The letter`@`is used if a needed prime exponent is zero in the 26-base system. In the case that this identifier is still ambigious, it is appended by a lower case letter which serves as a counter.

`depth`indicates the minimal number of successive primes (starting with 2) whose exponents must be given to reconstruct the MPN straight forward (without knowing its abundancy).`dpf`indicates the number of different prime factors.`tpf`indicates the number of total prime factors.`date`gives the year of the first (or independent) discovery in the form`YYYY-MM-DD`as long as month and day (and year) is known.`name`gives the name of the discoverer. Like in the date field, multiple independent discoveries are separated by commas.`number`is given in its prime factorization of the form:`base1^expo1.base2^expo2.base3^expo3. ....`where the`basei`indicates a prime of an ascending list of primes and the`expoi`indicates the corresponding nonzero exponent; an exponent value of 1 is omitted together with its preceding`^`. In the case of the larger 2-perfect numbers, the odd prime (`2^??-1`) in the factorization is given as`M??`to save space with`??`giving the**Mersenne Prime exponent**as decimal number.

**Search this database of 5932 MPNs**

Each search-pattern matches literally, except the wild-characters *,?,!,$ which mean:

* -- arbitrary, maybe empty, sequence of characters

? -- exactly one character

! -- begin of record

$ -- end of record.

Hint: orient on the field separator | or the comma in each record to get easier what you want.

And a further list of multiply perfect numbers sorted only by their factorizations (built from the 5932 MPNs of the master list and gziped 226 kB).
It has each line cut down to 120 columns and no discovery information or comments are given. Rich's
out-dated indentifier is omitted, but a log_{10}(MPN) is given for traditional reasons.

Finally, here are the (email-extracted and edited to put it syntactically in standard form)
0 claimed new (since the database update) found MPNs which I irregularly append (last change: 2023-01-12).
To verify these numbers, three steps must be taken: 1) verify for each number *n*
given in its prime factorization that all factors are really prime numbers, 2) compute *σ(n)* which envolves the factorization of large numbers and then check *n*'s claimed abundancy 3) and lastly test whether *n* is really new.

Moreover there is the case that for a fixed (small) prime

A = 19^2.127^1.151^0.911^0 B = 19^4.127^0.151^1.911^1 size(A,B) = 4 , k = 4,5,6,7,8,9 [occurs 257 times for the known MPNs] C = 23^1.37^3.73^1.79^0.137^2 D = 23^2.37^1.73^0.79^1.137^1 size(C,D) = 5 , k = 6,7,8,9 [occurs 21 times for the known MPNs] E = 13^1.31^2.61^1.83^1.97^0.331^1 F = 13^2.31^1.61^2.83^0.97^1.331^0 size(E,F) = 6 , k = 6,7,8 [occurs 31 times for the known MPNs] G = 3^10.107^1.137^0.547^0.1093^0.3851^1 H = 3^6.107^0.137^1.547^1.1093^1.3851^0 size(G,H) = 6 , k = 4,5,6 [occurs 31 times for the known MPNs] I = 5^0 J = 5^1 size(I,J) = 1 , k = 5 [occurs 20 times for the known MPNs] K = 7^5.17^1.37^1.43^1.67^0.307^0.1063^0 L = 7^8.17^2.37^2.43^0.67^1.307^1.1063^1 size(K,L) = 7 , k = 7 [occurs 9 times for the known MPNs] M = 7^4.13^4.31^1.43^1.61^1.79^0.97^0.157^0.631^0.30941^1 N = 7^9.13^5.31^0.43^2.61^2.79^2.97^1.157^1.631^1.30941^0 size(M,N) = 10 , k = 5,6,7,8 [occurs 4 times for the known MPNs] P = 2^38.53^1.59^0.157^0.229^1.8191^1.43331^0.121369^1.3033169^0.715827883^0.2147483647^0 Q = 2^61.53^0.59^1.157^1.229^0.8191^0.43331^1.121369^0.3033169^1.715827883^1.2147483647^1 size(P,Q) = 11 , k = 5,6 [occurs 9 times for the known MPNs] R = 13^14.139^0.157^1.181^1.191^2.199^1.229^1.397^1.827^0.1163^1.4651^1.8269^0.14197^0.28393^0.30941^1.40493^1.161971^1 S = 13^11.139^1.157^2.181^2.191^1.199^0.229^2.397^0.827^1.1163^0.4651^0.8269^1.14197^1.28393^1.30941^0.40493^0.161971^0 size(R,S) = 17 , k = 9,10 [occurs 2 times for the known MPNs] T = 3^4.11^3.13^1 U = 3^5.11^1.13^2 size(T,U) = 3 , k = 6 [occurs 8 times for the known MPNs] V = 5^2.7^2.11^1.31^1.71^0 W = 5^4.7^3.11^2.31^0.71^1 size(V,W) = 5 , k = 6 [occurs 8 times for the known MPNs]They are as more important, as fewer primes are involved in such a substitution. As larger the smallest prime in such a pair is, as more usefull it is, too. On the other hand, as smaller the quotient of the number of different primes in

Abundancy | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|

# known mpns | 6 | 36 | 65 | 245 | 516 | 1136 | 2164 | 1710 | 2 |

# different subst | 0 | 30 | 37 | 283 | 59 | 6 | 2 | 0 | 0 |

period / year | Antiquity | Middle Ages | < 1910 | < 1915 | < 1930 | < 1955 | < 1980 | < 1990 | 1991 | 1992 | 1993 | 1994 | 1995 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

known MPNs | 5 | 35 | 47 | 235 | 347 | 556 | 596 | 611 | 713 | 1179 | 1821 | 1983 | 2101 |

increament | 5 | 30 | 12 | 188 | 112 | 209 | 40 | 15 | 102 | 466 | 642 | 162 | 118 |

period | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

known MPNs | 2335 | 3051 | 3259 | 3464 | 4501 | 4997 | 5040 | 5050 | 5147 | 5189 | 5222 | 5245 | 5271 | 5287 | 5301 | 5303 | 5307 | 5311 |

increament | 234 | 716 | 208 | 205 | 1037 | 496 | 43 | 10 | 97 | 42 | 33 | 23 | 26 | 16 | 14 | 2 | 4 | 4 |

period / year | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
---|---|---|---|---|---|---|---|---|---|

known MPNs | 5311 | 5311 | 5312 | 5314 | 5315 | 5315 | 5315 | 5772 | 5932 |

increament | 0 | 0 | 1 | 2 | 1 | 0 | 0 | 457 | 160 |

The value in the row

Now a list of the discoverer names follows according to Rich Schroeppel (report him, if you think you aren't credited sufficiently):

Discoverer | count | period | Discoverer | count | period |
---|---|---|---|---|---|

Armengaud&Woltman&et.al. | 1* | 1996 | Brown | 78 | 1954 |

Cameron&Woltman&Kurowski&et.al. | 1* | 2001 | Carmichael | 98 | 1907-1911 |

Cataldi | 2* | 1588 | Clarkson&Woltman&Kurowski&et.al. | 1* | 1998 |

codex-Lat.-Monac.- | 1* | 1456 | Colquitt&Welsh | 1* | 1988 |

Cooper&Boone&Woltman&Kurowski&et.al. | 2* | 2005-2006 | Cooper&Woltman&Kurowski&et.al. | 1* | 2013 |

Cooper&Woltman&Kurowski&Blosser&et.al. | 1* | 2016 | Cunningham | 3 | 1902 |

Descartes | 9 | 1638-1639 | Dokshitzer&Vasyunin | 1 | 1993 |

Euclid | 4* | -275 | Euler | 1* | 1772 |

Elvenich&Woltman&Kurowski&et.al. | 1* | 2008 | Fermat | 11? | 1636-1643 |

Findley&Woltman&Kurowski&et.al. | 1* | 2004 | Flammenkamp | 137 | 1993-2000 |

Flammenkamp&Woltman | 344 | 2000-2001 | Franqui&Garcia | 135 | 1953-1954 |

Freniclei | 3 | 1638 | Gage&Slowinski | 3* | 1992-1996 |

Gillies | 3* | 1963 | Gretton | 123 | 1990-1993 |

Gretton&Schroeppel | 5 | 1991-1992 | Hajratwala&Woltman&Kurowski&et.al. | 1* | 1999 |

Harrison&Moxham | 3 | 2000 | Helenius | 1261 | 1992-1997 |

Helenius&Schroeppel | 16 | 1993 | Hurwitz&Selfridge | 2* | 1961 |

in-Mersenne-epoch | 4 | 1639-1643 | Jumeau | 1 | 1638 |

Kapek&Moxham | 1 | 2000 | Laroche&Woltman&Blosser&et.al. | 1* | 2018 |

Lehmer | 5 | 1901 | Lucas | 1* | 1876 |

Martinson&Moxham | 1 | 2000 | Mason | 89 | 1911 |

Moxham | 1006 | 1995-2001 | Nelson&Slowinski | 1* | 1979 |

Nickel&Noll | 1* | 1978 | Noll | 1* | 1979 |

Nowak&Woltman&Kurowski&et.al. | 1* | 2005 | Pace&Woltman&Kurowski&Blosser&et.al. | 1* | 2017 |

Perrier&Woltman | 38 | 2000 | Pervouchine | 1* | 1883 |

Poulet | 145 | 1929,1954 | Powers | 2* | 1911-1914 |

Pythagoras | 1* | -500 | Recorde | 1 | 1557 |

Riesel | 1* | 1957 | Roberts&Moxham | 2 | 1997 |

Robinson | 5* | 1952 | Schroeppel | 7 | 1983-1991 |

Shafer&Woltman&Kurowski&et.al. | 1* | 2003 | Slowinski | 3* | 1982-1985 |

Smith&Woltman&Kurowski&et.al. | 1* | 2008 | Sorli&Moxham | 27 | 1997-1998 |

Sorli&Woltman | 9 | 2000-2001 | Spence&Woltman&et.al. | 1* | 1997 |

Strindmo&Woltman&Kurowski&et.al. | 1* | 2009 | Toshihiro | 1 | 2017 |

Tuckerman | 1* | 1971 | Whiteside&Moxham | 2 | 1999 |

Woltman | 2416 | 1997-2013,2021-2022 | Yoshitake | 17 | 1974-1993 |

Abundancy | First discovered MPN | Is it smallest? | Date | Discoverer |
---|---|---|---|---|

1 | 1 | yes | ancient | - |

2 | 6 | yes | ancient | - |

3 | 120 | yes | ancient | - |

4 | 30240 | yes | ~ 1638 | R. Descartes |

5 | 14182439040 | yes | ~ 1638 | R. Descartes |

6 | 34111227434420791224041472000 | no | 1643 | P. Fermat |

7 | 6.9545266398342727... *10^70 | no | 1902 | A. J. C. Cunningham |

8 | 2.34111439263306338... *10^161 | no | 1929 | P. Poulet |

9 | 7.9842491755534198... *10^465 | no | 1992-04-15 | F. W. Helenius |

10 | 2.86879876441793479... *10^923 | no | 1997-05-13 | R. M. Sorli |

11 | 2.51850413483992918... *10^1906 | no | 2001-03-13 | G. F. Woltman |

- Abundancy 1:

1 - Abundancy 2:

2 3 - Abundancy 3:

2^3 3 5 - Abundancy 4:

2^5 3^3 5 7 - Abundancy 5:

2^7 3^4 5 7 11^2 17 19 - Abundancy 6:

2^15 3^5 5^2 7^2 11 13 17 19 31 43 257 - Abundancy 7:

2^32 3^11 5^4 7^5 11^2 13^2 17 19^3 23 31 37 43 61 71 73 89 181 2141 599479 - Abundancy 8:

2^62 3^15 5^9 7^7 11^3 13^3 17^2 19 23 29 31^2 37 41 43 53 61^2 71^2 73 83 89 97^2 127 193 283 307 317 331 337 487 521^2 601 1201 1279 2557 3169 5113 92737 649657 - Abundancy 9:

2^104 3^43 5^9 7^12 11^6 13^4 17 19^4 23^2 29 31^4 37^3 41^2 43^2 47^2 53 59 61 67 71^3 73 79^2 83 89 97 103^2 107 127 131^2 137^2 151^2 191 211 241 331 ... - Abundancy 10:

2^175 3^69 5^29 7^18 11^19 13^8 17^9 19^7 23^9 29^3 31^8 37^2 41^4 43^4 47^4 53^3 59 61^5 67^4 71^4 73^2 79 83 89 97 101^3 103^2 107^2 109 113 127^2 ... - Abundancy 11:

2^413.3^145 5^73 7^49 11^27 13^22 17^11 19^13 23^10 29^9 31^8 37^4 41^8 43^4 47^6 53 59 61^6 67^6 71^4 73^4 79^4 83^2 89^2 97^2 101^3 103^2 107^2 ...

Abund | Smallest value | Largest value | Lowest 2-power | Highest 2-power | Largest eff. expo. | Fewest factors * | Most factors * |
---|---|---|---|---|---|---|---|

2 | 6 (0.5831981) | 10^49724093.6 (18.5560326) | 1 (0.5831981) | 82589932 (18.5560326) | 0 (0.5831981) | 2 (0.5831981) | 82589933 (18.5560326) |

3 | 120 (1.5660066) | 10^10.469 (3.2049844) | 3 (1.5660066) | 14 (3.2049844) | 7 (3.2049844) | 5 (1.5660066) | 19 (3.2049844) |

4 | 30240 (2.3337853) | 10^45.204 (4.6452114) | 2 (2.6806218) | 37 (4.5312242) | 17 (4.5312242) | 8 (2.3415138) | 57 (4.6365489) |

5 | 10^10.148 (3.1516786) | 10^100.276 (5.4419551) | 7 (3.1516786) | 61 (5.0442101) | 45 (5.1777077) | 17 (3.1516786) | 103 (5.3046236) |

6 | 10^20.189 (3.8391453) | 10^192.162 (6.0923690) | 15 (4.0050961) | 92 (6.0923690) | 79 (6.0362173) | 31 (3.8391453) | 164 (6.0923690) |

7 | 10^56.150 (4.8620623) | 10^312.074 (6.5772722) | 29 (5.2987825) | 177 (6.5408889) | 108 (6.4271649) | 71 (4.8620623) | 307 (6.5528603) |

8 | 10^132.917 (5.7237604) | 10^613.291 (7.2528723) | 47 (5.7578915) | 253 (7.1699823) | 156 (7.2528723) | 137 (5.7237604) | 466 (7.1485402) |

9 | 10^286.749 (6.4926404) | 10^1165.883 (7.8952666) | 99 (6.6259527) | 380 (7.8522986) | 283 (7.7478296) | 257 (6.4926404) | 747 (7.8522986) |

10 | 10^638.652 (7.2933919) | 10^1877.645 (8.3718042) | 175 (7.2933919) | 534 (8.3718042) | 434 (8.2500454) | 492 (7.2933919) | 1172 (8.3718042) |

11 | 10^1738.495 (8.2948075) | 10^1906.401 (8.3870050) | 413 (8.2948075) | 468 (8.3870050) | 469 (8.3870050) | 1088 (8.2948075) | 1139 (8.3870050) |

The identifiers for each MPN are given in parenthesis.

(*) The word factors means prime-factors.

Discovery dates of proper Multiply Perfect Numbers

Abund | First MPN | Smallest MPN | Largest MPN | Latest MPN |
---|---|---|---|---|

3 | ancient (1.5660066) | ancient (1.5660066) | 1643 (3.2049844) | 1643 (3.2049844) |

4 | 1638 (2.3337853) | 1638 (2.3337853) | 1911 (4.6452114) | 1929 (4.3351682) |

5 | 1638 (3.1516786) | 1638 (3.1516786) | 1911 (5.4419551) | 1990 (5.1744360) |

6 | 1643 (4.1850902) | 1907 (3.8391453) | 1992 (6.0923690) | 1993-05-?? (5.6720844) |

7 | 1902 (5.0944883) | 1911 (4.8620623) | 1993 (6.5772722) | 1994-01-09 (5.9403364) |

8 | 1929 (5.9177287) | 1990 (5.7237604) | 1996-04-23 (7.2528723) | 2022-05-22 (6.5244184) |

9 | 1992-04-15 (6.9780083) | 1995-12-06 (6.4926404) | 2001-01-11 (7.8952666) | 2022-06-05 (7.1993123) |

10 | 1997-05-13 (7.6621574) | 2013-01-03 (7.2933919) | 2001-01-28 (8.3718042) | 2022-06-11 (7.8125224) |

11 | 2001-03-13 (8.3870050) | 2022-09-07 (8.2948075) | 2001-03-13 (8.3870050) | 2022-09-07 (8.2948075) |

- The only odd multiply perfect number is 1. Consequently each (2-fold) perfect number is even (claimed already in the Middle Ages).
- There are infinitely many perfect, i.e. 2-fold multiply perfect, numbers.
- For each fixed abundancy > 2, there are only finitely many multiply perfect numbers.
- For each fixed prime power, there is at least one MPN which has exactly this prime power in its prime factorization (lowest two-power-exponents for to-be-discovered-MPNs are 331, 336, 347, ... and lowest three-power exponents are 127,134, ... ).

If the number of MPNs up to a given limit x is proportional to ln(x), then the density of the MPNs should be proportional to 1/x.
And a further picture to visualize the density of the known MPNs.
In first approximation it may be a Poisson-distribution.
You guess that surely there are missing MPNs at last at the currently known 4000th MPN, but also highly probable earlier.
Finally a best linear fit for the smallest 3600 known MPNs except the first 100 MPNs to avoid 'start-effects' was done on a logarithmic scale. For these 3500 numbers this least-quadratic error approximation results a correlation coefficient of 0.99815 with *# MPN ~ 2.2328ln(x)-48.3655*.

Richard Schroeppel stated
that he constructed all MPNs < 10^{70} in the eighties of the twenty-century (but probably only 1990),
i.e. he proved that there exist exactly 258 MPNs smaller than 10^{70} -- and besides showed that there are no odd MPNs < 10^{90} except 1.
In March 2008 Achim Flammenkamp finally constructed all MPNs < e^{350} by an exhaustive tree-search, i.e. proved that there are no further but the 730 known MPNs.
Here is a small paper presenting the theoretical and practical background to this computation.

And here is a false color diagram for the distribution of these MPNs

Let us introduce Fred Helenius' notion of the **effective exponent**.
The prime *p=2* is special compared to other primes, because
*p-1* has only the trivial divisor *1*.
A consequence of this fact is, only if *p* equals *2*, then numbers of the form *p ^{n}-1* could be prime.
Assume a fixed MPN

The two-power exponent

Finally, for a given two-power exponent

- R. D. Carmichael & T. E. Mason, Notes on Multiply Perfect Numbers, Including a Table of 204 New Ones and the 47 Others Previously Published, Proc. Indiana Academy of Science, 1911 p257-270.
- Leonard Eugene Dickson, History of the Theory of Numbers, 1919, v.1 p33-38.
- Paul Poulet, La Chasse Aux Nombres, Fascicule I, Bruxelles, 1929, p9-27.
- Benito Franqui & Mariano Garcia, Some New Multiply Perfect Numbers, American Math Monthly 1953 p459-462.
- Alan L. Brown, Multiperfect Numbers, Scripta Mathematica 1954 p103-106.
- Benito Franqui & Mariano Garcia, 57 New Multiply Perfect Numbers, Scripta Mathematica 1954 p169-171.
- Alan L. Brown, Multiperfect Numbers - Cousins of the Perfect Numbers - No. 1, Recreational Mathematics Magazine #14, Jan/Feb 1964.
- Motoji Yoshitake, Abundant Numbers, Sum of Whose Divisors are an Integer Times the Number, Sugaku Seminar, v.18 n.3 p50-55, 1979.
- private communications from M. Garcia, Stephen Gretton, M. Yoshitake, Fred Helenius, and Achim Flammenkamp.

- Fred Helenius's Multiperfect numbers
- Ron Sorli's Thesis:
*Algorithms in the Study of Multiperfect and Odd Perfect Numbers*2003, Sydney, Australia and its first 9 pages . - A gziped-tar archive containing the sigma-chain database and its C-sourcecode library (486kB) to access and maintain it (version 4.1d).
- Eric Weisstein's explanations of MPNs. .
- The haunt for improper Multiperfect Numbers :)

Please sent **any comments or questions** concerning this web page to:

Achim Flammenkamp

2023-01-21 15:26 UTC+1