The Multiply Perfect Numbers Page

Introduction

Let o(n) be the number theoretic function which denotes the sum of all divisors of a natural number n. If o(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 o(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=o(120)=o(2^3*3*5)=o(2^3)*o(3)*o(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 48 2013-01-2518.2006059 no, there are infinitely many oo
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 1134 2000-10-116.7584723 probably yes ~ 1140
9 2095 2013-01-107.2802453 no ~ 2200
10 1164 2013-01-037.2933919 no ~ 4500
11 1 2001-03-138.3870050 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 2014-01-01.
We have a total of 5311+0 (of which 5262 have an abundancy > 2) known and claimed MPNs until 2014-01-19.
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 5311 MPNs from 2014-01-01(gziped to 918 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 with all its additional informations in the form:
M|ln_ln|rich_id,deep|dpf,tpf|date|name|number|comment

Search this database of 5311 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 5311 MPNs of the master list and gziped 200 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) 0 claimed new (since the database update) found MPNs which I irregularly append (last change: 2014-01-20). 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 o(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 o(n*X)=k*n*X and o(n*Y)=k*n*Y, with gcd(n,X)=1=gcd(n,Y), i.e. they have no primes in common. Then yields: if n*X is a k-perfect number then n*Y is one, too. Such a pair (X,Y) is denoted as a substitution. To avoid trivial substitutions, one demands further that prime-powers whose prime occuring in both components have different exponents. Call the size of a substitution the number of different primes involved. 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
[occurs 1 time 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.

Historical Development of known MPNs

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
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
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
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 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
Tuckerman 1*1971 Whiteside&Moxham 1999
Woltman 1799 1997-2013 Yoshitake 17 1974-1993
The column 'count' counts the discoveries until 2013-12-31 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

The factorizations of smallest -- for k >= 9 "only" smallest known -- k-perfect number for fixed abundancy are:

Abund Smallest value Largest value Lowest 2-power Highest 2-power Largest eff. expo. Fewest factors * Most factors *
2 6 (0.5831981)10^34850339.2 (18.2006059)1 (0.5831981)57885160 (18.2006059)0 (0.5831981)2 (0.5831981)57885161 (18.2006059)
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 10^1906.401 (8.3870050)10^1906.401 (8.3870050)468 (8.3870050)468 (8.3870050)469 (8.3870050)1139 (8.3870050)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)2000-10-11 (6.7584723)
9 1992-04-15 (6.9780083)1995-12-06 (6.4926404)2001-01-11 (7.8952666)2013-01-10 (7.2802453)
10 1997-05-13 (7.6621574)2013-01-03 (7.2933919)2001-01-28 (8.3718042)2013-01-03 (7.2933919)
11 2001-03-13 (8.3870050)2001-03-13 (8.3870050)2001-03-13 (8.3870050)2001-03-13 (8.3870050)
Italic printed values perhaps change in the future.

Old conjectures

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

MPN-size-diagram

MPN-dist-diagram
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

MPN-two-power-diagram
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 o(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, o(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

Further Infos

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Achim Flammenkamp
2014-01-25 19:35 UTC+1