# The Multiply Perfect Numbers Page

## Introduction

Let σ(n) be the number theoretic function which denotes the sum of all divisors of a natural number n. If σ(n) is an integral multiply of n, then n is denoted as a multiply perfect number or k-fold perfect number (also called multiperfect number or pluperfect number). Call σ(n)/n abundancy (also called index or multiplicity) of n. A multiply perfect number is called proper if its abundancy is > 2. For example consider the divisors of the number 120: 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120=σ(120)=σ(2^3*3*5)=σ(2^3)*σ(3)*σ(5)=(1+2+4+8)*(1+3)*(1+5)=15*4*6=360=3*120.
Hence 120 is a 3-fold perfect number.

## Status

Abundancy Count When last number was discovered Which was last?Are all discovered?Estimated total number
1 1 - - yes and proved 1
2 51 2018-12-0718.5560326 no, there are infinitely many
3 6 <= 16433.2049844 yes 6
4 36 <= 19294.3351682 yes 36
5 65 <= 19905.1744360 yes 65
6 245 1993-05-??5.6720844 yes 245
7 516 1994-01-095.9403364 almost surely yes ~ 515
8 1136 2022-05-226.5244184 probably yes ~ 1140
9 2164 2022-06-057.1993123 no ~ 2200
10 1710 2022-06-117.8125224 no ~ 4500
11 2 2022-09-078.2948075 no ~ 10000
In column "Which was last?" the identifier ln(ln(MPN)) is given for those which were verfied by me. I checked these numbers only for those MPNs reported before 2023-01-01.
We have a total of 5932+0 (of which 5880 have an abundancy > 2) known and claimed MPNs until 2022-12-31.
It is extremly probable, that all proper MPNs with abundancy <= 7 are discovered.

## Data

Richard Schroeppel's archive of 2094 MPNs built 1995-12-13 .
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:
M|ln_ln|rich_id,depth|dpf,tpf|date|name|number|comment
Except of the last field, comment, all other fields are obligatory, but they can be empty, e.g. if the discovery date or person is unknown.
• | 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 logeloge 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.

Pattern to search.
output fields of records -

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 log10(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.

## Algebraic Transformations

Let σ(n*X)=k*n*X and σ(n*Y)=k*n*Y, with gcd(n,X)=1=gcd(n,Y), i.e. they have no primes in common. Such a pair (X,Y) is denoted as a substitution. To avoid trivial substitutions, one demands further that prime-powers whose primes occuring in both components have different exponents. This yields: Given integers n, X, Y with gcd(n,X)=1=gcd(n,Y). If n*X is a k-perfect number and X*σ(n*Y)=Y*σ(n*X) -- i.e. X and Y have the same "fractional abundancy" -- holds, then n*Y is a k-perfect number, too. Call the size of a substitution the number of different primes involved.
Moreover there is the case that for a fixed (small) prime k the number n is a k-perfect number and gcd(n,k)=1. Then the number k*n is k+1-perfect! I.e. if n has no prime factor k we can construct a further multiply perfect number of a by 1 increased multiplicity. Below are some components listed for pairs (A,B), (C,D), (E,F), (G,H), (I,J), (K,L), (M,N), (P,Q), (R,S), (T,U) and (V,W):
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 n to the size of the substitution becomes, as more worthless such a substitution becomes for this n. Therefore we demand arbitrarily that the size of a substitution must be less than the number of different primes in each of the two components. Sadly it looks like that substitutions occur only for mpns with small abundancy as the following table of counts for substitutions of size of at most 12 suggests:
 Abundancy # known mpns # different subst 3 4 5 6 7 8 9 10 11 6 36 65 245 516 1136 2164 1710 2 0 30 37 283 59 6 2 0 0
Then the number of (different) substitutions of size ≤ 12 is finite and probably all are already discovered.

## Historical Development of known MPNs

 period / year known MPNs increament Antiquity Middle Ages < 1910 < 1915 < 1930 < 1955 < 1980 < 1990 1991 1992 1993 1994 1995 5 35 47 235 347 556 596 611 713 1179 1821 1983 2101 5 30 12 188 112 209 40 15 102 466 642 162 118
 period known MPNs increament 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2335 3051 3259 3464 4501 4997 5040 5050 5147 5189 5222 5245 5271 5287 5301 5303 5307 5311 234 716 208 205 1037 496 43 10 97 42 33 23 26 16 14 2 4 4
 period / year known MPNs increament 2014 2015 2016 2017 2018 2019 2020 2021 2022 5311 5311 5312 5314 5315 5315 5315 5772 5932 0 0 1 2 1 0 0 457 160
The sign < in the row period means "up to and including" the given year.
The value in the row increament indicates the new known MPNs compared to the previous time-period.

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 1902
Descartes 1638-1639 Dokshitzer&Vasyunin 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 1638 Gage&Slowinski 3*1992-1996
Gillies 3*1963 Gretton 123 1990-1993
Gretton&Schroeppel 1991-1992 Hajratwala&Woltman&Kurowski&et.al. 1*1999
Harrison&Moxham 2000 Helenius 1261 1992-1997
Helenius&Schroeppel 16 1993 Hurwitz&Selfridge 2*1961
in-Mersenne-epoch 1639-1643 Jumeau 1638
Kapek&Moxham 2000 Laroche&Woltman&Blosser&et.al. 1*2018
Lehmer 1901 Lucas 1*1876
Martinson&Moxham 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&Woltman38 2000 Pervouchine 1*1883
Poulet 145 1929,1954 Powers 2*1911-1914
Pythagoras 1*-500 Recorde 1557
Riesel 1*1957 Roberts&Moxham 1997
Robinson 5*1952 Schroeppel 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 2000-2001 Spence&Woltman&et.al. 1*1997
Strindmo&Woltman&Kurowski&et.al. 1*2009 Toshihiro 2017
Tuckerman 1*1971 Whiteside&Moxham 1999
Woltman 2416 1997-2013,2021-2022 Yoshitake 17 1974-1993
The column 'count' counts the discoveries until 2022-12-31 and an asterik * indicates that this person has "only" discovered 2-perfect number(s).

Abundancy First discovered MPN Is it smallest? Date Discoverer
1 1 yes ancient -
2 6 yes ancient -
3 120 yesancient -
4 30240 yes~ 1638R. Descartes
5 14182439040 yes~ 1638R. Descartes
6 34111227434420791224041472000 no1643P. Fermat
7 6.9545266398342727... *10^70 no1902A. J. C. Cunningham
8 2.34111439263306338... *10^161 no1929P. Poulet
9 7.9842491755534198... *10^465 no1992-04-15F. W. Helenius
10 2.86879876441793479... *10^923 no1997-05-13R. M. Sorli
11 2.51850413483992918... *10^1906 no2001-03-13G. F. Woltman

## Various Records until January 2023

The factorizations of smallest -- for k >= 9 "only" smallest known -- k-perfect number for fixed abundancy are:
• 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)
Italic printed values perhaps change in the future.
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)
Italic printed values perhaps change in the future.

## Old conjectures

• 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, ... ).

## Distribution of the 5311 (until 2013-12-31) known Multiply Perfect Numbers

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 < 1070 in the eighties of the twenty-century (but probably only 1990), i.e. he proved that there exist exactly 258 MPNs smaller than 1070 -- and besides showed that there are no odd MPNs < 1090 except 1. In March 2008 Achim Flammenkamp finally constructed all MPNs < e350 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.

### The Two-Power exponent of a MPN

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 pn-1 could be prime. Assume a fixed MPN m has in its prime factorization the two-power exponent k. This means that σ(m) has a factor 2k+1-1. Depending on k this factor may have some prime divisors of the form 2n-1. Because we are considering a MPN, σ(2n-1) = 2n must be also a factor of this MPN itself, if 2n-1 occurs exactly onetime! Hence the original prime-power 2k may produce immediately further two-powers. Or put it in other words: only the `remainder' of this exponent k must be produced by other prime-powers. So, the exponent is effectively reduced in consideration to the to-be-generated two-power factors by other prime-powers. Primes of the form 2n-1 are called Mersenne Primes and are known up to high values of n. Thus we have to check which Mersenne Prime 2n-1 divide a given 2k+1-1 exactly onetimes. Subtracting k+1 by such Mersenne Prime exponents n gives the effective exponent of k. There is a small blemish in this model: small primes of the form q=2n-1 may as well be produced by other prime-powers (than two-powers) of a MPN, such that the exponent of q is not 1 in the factorization of the MPN, but larger. Hence we have probably overestimated the correction of the two-power exponent a bit.
The two-power exponent k of a (2-fold) perfect number has always an effective exponent of 0. For such an effective exponent of 0, there seems to exist only the corresponding (2-fold) MPN except in the case k equals 2. If k = in-1 with n the exponent of a Mersenne Prime and i a small number, the effective exponent is at most n(i-1), roughly 1-1/i of the given exponent k. Such exponents k are typically the largest two-power exponents for which a MPN for a fixed proper abundancy exists: 61=2*31-1, 92=3*31-1, 177=2*89-1, 253=2*127-1, 320=3*107-1,380=3*127-1.
Finally, for a given two-power exponent k the effort to compute a MPN with an even factor of 2k seems more related to its effective exponent than to k --- this is heuristically convincing and seems likely to be the reason this quantity was invented --- and lastly the distribution of the prime-exponents of a `typical' MPN seems proportional to the effective exponent than to the two-power exponent.

## References collected by Rich Schroeppel

• 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.