### Abundancy : some resources ( preliminary version 4 )

Abundancy is defined as the ratio of the multiplicative sum-of-divisors
function to the integer itself .

It can be particularly difficult to locate resources on abundancy
because so many authors have given so many different names to abundancy
and because many other authors have used the ratio , but never bothered
to give it any name at all .
Some of the names have been  :  abundancy index , index ,
abundancy ratio , ratio , m , P , abundance , k-ply , r , SIGMA ,
sigma-sub-1 , index of perfection , perfectness , k , h , S , class ,
k-fold , multiplicity , I , kinship , friendly pair checker ,
relative abundance , rho , relative sum of divisors function ,
perfection quotient , p , characteristic ratio , u-sub-n , etc.

A personal note :
I think abundancy is one of the pre-eminent number-theoretic functions ,
and think it should have a consistent name .
Also that that name , in English , should be abundancy .
Here abund is used as an abbreviation for abundancy .

- - - - -

Leonhard Euler
I. Arithmetica , III. De numeris amicabilibus , Section 8 [unnamed]
1. Arithmetic , 3. On amicable numbers , Section 8
Opera postuma mathematica et physica: anno MDCCCXLIV detecta , Volume 1
Posthumous Works on Math and Physics: revealed 1844 , Volume 1
1862
p. 88
http://www.archive.org/details/leonhardieuleri00petrgoog
p. n105
http://math.fau.edu/yiu/eulernotes99.pdf    ( and .ps )
p. 106
uses abundancy , unnamed , to prove every even perfect is of Euclid's type

R. D. Carmichael
A table of multiply perfect numbers
Bull. Amer. Math. Soc. , 1907 , V13#8pp383-386
http://www.ams.org/bull/1907-13-08/S0002-9904-1907-01483-0/S0002-9904-1907-01483-0.pdf
refers to abundancy of multiperfects as "multiplicity"

R. D. Carmichael , T. E. Mason
Note on Multiply Perfect Numbers, Including a Table of
204 New Ones and the 47 Others Previously Published
Proc. of the Indiana Academy of Science , 1911 , pp257-270
incl. table of primitive friendly pairs , not so labelled , and
Th. from Descartes akin to exclusive multiplication

Leonard Eugene Dickson
Finiteness of the odd perfect and primitive abundant numbers
with  n  distinct prime factors
American Journal of Mathematics , 1913 Oct , V35#4pp413-422
http://www.jstor.org/stable/2370405
refers to abundancy as "P"

Leonard Eugene Dickson
Even abundant numbers
American Journal of Mathematics , 1913 Oct , V35#4pp423-426
http://www.jstor.org/stable/2370406

Leonard Eugene Dickson
History of the Theory of Numbers , Volume I , Divisibility and Primality
1919 , Carnegie Institution of Washington
2005 , Dover Publications , ISBN 9780486442327
http://www.amazon.com/dp/0486442322/
pp. 3-38 , IV , 26 , 28 , 19 , 281, 291 , especially pp. 53 , 35
on p. 19 , quoting Euler , uses abundancy to prove every even perfect is
of Euclid's type
refers to abundancy of multiperfects as "multiplicity"
p. 35 refers to use of sum-of-aliquot-divisors rather than sum-of-divisors
also one of the references on p. 53
"This history aims to give an adequate account of the entire literature
of the theory of numbers."
For some purposes ,
Richard K. Guy
Unsolved Problems in Number Theory
q.v. , may be seen as a successor to this history .
errata : p 200

R. D. Carmichael
Review of :
History of the Theory of Numbers , Volume I , Divisibility and Primality
American Mathematical Monthly , 1919 Nov , V26#9pp396-403
http://www.jstor.org/stable/2971917
"To give an adequate account of the entire history of the theory of
numbers is an undertaking of enormous magnitude; and it is carried
through in this work with a marvelous success in the presence of which
one must pause in admiration. ... this history will be
indispensable to all investgators in the theory of numbers. ...
It is a piece of work for which one cannot find a parallel in the whole
of scientific history."
refers to abundancy of multiperfects as "multiplicity"
mentions minPfp = { 19^2*127 ; 19^4*151*911 } = { 45847 ; 17927087081 }

Leonard Eugene Dickson
Perfect and Amicable Numbers
Scientific Monthly , AAAS , 1921 Apr , V12#4pp349-354
http://www.jstor.org/stable/6610
abundancy entirely implicit
3           p
"We shall write 2 for 8 and 2 for the product of p factors 2. ...
if q is any prime and n is any whole number not divisible by q,
q-1
then n   - 1 is divisible by q.  This result, which is the basis
of the modern theory of numbers, is known as Fermat's theorem. ...
It is not surprising that Mersenne's guesses were erroneous,
but it is quite surprising that his errors have been detected"

Thomas E. Mason
On Amicable Numbers and Their Generalizations
American Mathematical Monthly , 1921 May , V28#5pp195-200
http://www.jstor.org/stable/2973750
refers to abundancy of multiperfects as "multiplicity" and "k"
"The methods of finding amicable number sets are very largely those of
trial.  Experience in working with such numbers will suggest the likely
numbers to try, but there is no sure guide yet known. ... Any systematic
search for amicable numbers among the large numbers will furnish a vast
amount of work."

According to
p. 7 , items above this point in this document are free of copyright
under US law .
Free online full text is available for many of these .

Paul Poulet
La Chasse Aux Nombres
Fascicule I , Bruxelles , Stevens Freres , 1929 , pp9-27
Fascicule I , Bruxelles , Stevens Freres , 1929 , pp9-27

S. S. Pillai
On sigma  (n) and phi(n)  =  On sigma-sub-1 (n) and phi(n)
-1
Proceedings of the Indian Academy of Sciences, Section A , 1943 Mar
V17#3pp67-70  or  pp70-
first(?) occurrence of abundancy , the concept , in a title

L. Alaoglu , Erd"os P'al = Paul Erdos
On Highly Composite and Similar Numbers
Transactions of the American Math. Society , 1944 Nov , V56#3pp448-469
http://www.jstor.org/stable/1990319
http://www.renyi.hu/~p_erdos/1944-03.pdf
superabundants = abundancy champions
a keystone paper

Oystein Ore
Number Theory and Its History
1948
1988 , Dover Publications , ISBN 0486656209

Harold N. Shapiro
Note on a theorem of Dickson
Bull. Amer. Math. Soc. , 1949 , V55#4pp450-452
http://projecteuclid.org/euclid.bams/1183513752
http://www.ams.org/bull/1949-55-04/S0002-9904-1949-09238-8/S0002-9904-1949-09238-8.pdf
uses ||

Benito Franqui , Mariano Garcia
Some new multiply perfect numbers
Amer. Math. Monthly , 1953 Aug , V60#7pp459-462
http://www.jstor.org/stable/2308408
features minPfp

Alan L. Brown
Multiperfect numbers
Scripta Mathematica , 1954 , pp103-106

Benito Franqui , Mariano Garcia
57 new multiply perfect numbers
Scripta Mathematica , 1954 , pp169-171

Erd"os P'al = Paul Erdos
Remarks on number theory, II ; Some problems on the sigma function
Acta Arithmetica , V5 , 1959 , pp171-177
http://www.renyi.hu/~p_erdos/1959-21.pdf
a keystone paper

Erd"os P'al = Paul Erdos
On the distribution of numbers of the form  sigma (n)/n  and on some
related questions
Pacific J. Math. , 1974 , V52#1pp59-65
http://www.renyi.hu/~p_erdos/1974-19.pdf
"As far as I know it has never been proved that for a suitable
alpha  the number of solutions of  sigma(n)/n = alpha  is infinite
-- or even unbounded in  alpha."

Erd"os P'al = Paul Erdos , Jean-Louis Nicolas
R'epartition des Nombres Superabondants
Repartition des Nombres Superabondants
Bull. Soc. math. France , 1975 , V103#1pp65-90
Bulletin de la Soci'et'e Math'ematique de France
Bulletin de la Societe Mathematique de France
http://www.renyi.hu/~p_erdos/1975-37.pdf
distribution of superabundants

Claude W. Anderson , Dean Hickerson
Amer. Math. Monthly , 1975 Mar , V82#3p307
[ vide infra ]
* proposed as unsolved

Claude W. Anderson , Neal Felsinger
Advanced Problem 5949 : Density of  sigma(n)/n
Amer. Math. Monthly , 1975 May , V82#5pp536-538
http://www.jstor.org/stable/2319765

News and Notices
Personal Items
Amer. Math. Monthly , 1975 Aug-Sep , V82#7p780
re : Claude W. Anderson
http://www.jstor.org/stable/2318757

Carl Pomerance               _                 _
On the congruences sigma (n) = a (mod n) and n = a (mod phi(n))
Acta Arithmetica , 1975 , V26pp265-272
http://math.dartmouth.edu/~carlp/PDF/paper7.pdf

Carl Pomerance
On multiply perfect numbers with a special property
Pacific J. Math. , 1975 , V57pp511-517
http://math.dartmouth.edu/~carlp/PDF/paper11.pdf

Claude W. Anderson , Dean Hickerson , M. G. Greening
Advanced Problem 6020 : Friendly Integers
Amer. Math. Monthly , 1977 Jan , V84#1pp65-66
http://www.jstor.org/stable/2318325
comments in lieu of a solution

Carl Pomerance
Multiply Perfect Numbers, Mersenne Primes and Effective Computability
Mathematische Annalen , 1977 Oct , V226#3pp195-206
http://math.dartmouth.edu/~carlp/PDF/paper13.pdf
notes that abund ( n ) = 5/3 yields an OPN
notes Steuerwald : 36 /| 3-perfect
in memory of Prof. C. W. Anderson

Erd"os P'al = Paul Erdos , C. W. Anderson , Robert E. Shafer
American Mathematical Monthly , 1977 Oct , V84#8pp662-663
http://www.jstor.org/stable/2321037
notes untimely passing of Prof. Claude W. Anderson

David G. Kendall
The Scale of Perfection
Journal of Applied Probability , Vol. 19 , Essays in Statistical Science (1982) , pp. 125-138
http://www.jstor.org/stable/3213555
abundancy  =  2 * "Kendall's scale of perfection"

Guy Robin
Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann
Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann
J. Math. Pures Appl. , 1984 , V63pp187-213

Richard Laatsch
Measuring the Abundancy of Integers
Mathematics Magazine , 1986 Apr , V59#2pp84-92
http://www.jstor.org/stable/2690424

Richard P. Brent , Graeme L. Cohen , H. J. J. Te Riele
Improved Techniques for Lower Bounds for Odd Perfect Numbers
Mathematics of Computation, 1991 Oct , V57#196pp857-868
http://www.jstor.org/stable/2938723

Walter Nissen
Exponential Prime Power Representation
1995 May 23
http://upforthecount.com/math/epprep.html

Kevin S. Brown
The Distribution of Perfection
1995 May 27
http://www.mathpages.com/home/kmath223.htm

Chris E. Thompson
re: The Distribution of Perfection
1995 May 29

Peter Hagis Jr. , Graeme L. Cohen
Every odd perfect number has a prime factor which exceeds 10^6
Math. Comp. , 1998 , V67#223pp1323-1330
http://www.ams.org/mcom/1998-67-223/S0025-5718-98-00982-X/S0025-5718-98-00982-X.pdf

Richard K. Guy
Nothing's New in Number Theory
Amer. Math. Monthly , 1998 Dec , V105#10pp951-954
http://www.jstor.org/stable/2589289

James G. Merickel , John P. Robertson
Problem 10617
Divisors of Sums of Divisors
Amer. Math. Monthly , 1999 Aug-Sep , V106#7p693
http://www.jstor.org/stable/2589515

Dean Hickerson
Re: friendly number
2000 Jan 31

Paul A. Weiner
The Abundancy Ratio, a Measure of Perfection
Mathematics Magazine , 2000 Oct , V73#4pp307-310
http://www.jstor.org/stable/2690980

Steve Pettigrew
Sur la distribution de nombres sp'eciaux cons'ecutifs
Sur la distribution de nombres speciaux consecutifs
On the occurrence of certain consecutive naturals
m'emoire , Universit'e Laval
memoire , Universite Laval
2000 Nov
abundant naturals , powerful naturals , table of abundancies

Can You Find (CYF)
http://www.shyamsundergupta.com/canyoufind.htm
2001 Dec 23

Jeffrey C. Lagarias
An Elementary Problem Equivalent to the Riemann Hypothesis
American Mathematical Monthly , 2002 Jun , V109#6pp534-543
http://arxiv.org/abs/math.NT/0008177/
The Riemann hypothesis is true iff , for  n > 5040 ,
abund ( n )  <=  e^gamma * loge ( loge ( n ) ) .
a remarkable paper

Judy A. Holdener
A Theorem of Touchard on the Form of Odd Perfect Numbers
Amer. Math. Monthly , 2002 Aug , V109#7p661-663
http://www.jstor.org/stable/3072433

Dean Hickerson
Re: friendly/solitary numbers [was: typos]
2002 Sep 19
seqfan mailing list , courtesy Olivier Gerard ( G'erard )

Richard F. Ryan
Results concerning uniqueness for
{\sigma (x)/x = \sigma (p^n q^m )/(p^n q^m)}  and related topics
International Math. J. , 2002 , V2#5pp497-514

Ronald M. Sorli
Algorithms in the Study of Multiperfect and Odd Perfect Numbers
2003 Mar 14
http://hdl.handle.net/2100/275

Slava Bezverkhnyev
Perfect Numbers and Abundancy Ratio
2003 Apr 3
http://www.slavab.com/Files/project.pdf

Mariano Garc'ia = Mariano Garcia , Jan Munch Pedersen , Herman te Riele
Amicable Pairs, a Survey
2003
http://www.cwi.nl/ftp/CWIreports/MAS/MAS-R0307.pdf
See "isotopic" on p. 5 , importantly establishes link to amicables ; and
on p. 16 , also
1 / abund  in equation (4.2) on p. 7 and on p. 8 and (4.4) on p. 9

Richard F. Ryan
A Simpler Dense Proof regarding the Abundancy Index
Mathematics Magazine , 2003 Oct , V76#4p299-301
http://www.jstor.org/stable/3219086

D. E. Iannucci , R. M. Sorli
On the Total Number of Prime Factors of an Odd Perfect Number
Math. Comp. , 2003 Oct , V72#244pp2077-2084
http://www.jstor.org/stable/4100039
http://www.ams.org/mcom/2003-72-244/S0025-5718-03-01522-9/home.html
http://www.ams.org/mcom/2003-72-244/S0025-5718-03-01522-9/S0025-5718-03-01522-9.pdf

Steven R. Finch
Multiples and Divisors
2004 Jan 27
http://algo.inria.fr/csolve/mldv.pdf

Walter Nissen
Primitive Friendly Integers and Exclusive Multiples
2004 Jul 02
http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0407&L=nmbrthry&F=&S=&P=51
http://upforthecount.com/math/mandrill.html

Richard K. Guy
Unsolved Problems in Number Theory , Sections B1 , B2 , B11
3e , 2004 , Springer-Verlag , ISBN 0387208607
For some purposes ,
Leonard Eugene Dickson
History of the Theory of Numbers
may be seen as a predecessor to this work .
a central work

Charles R. Greathouse IV
Bounding the Factors of Odd Perfect Numbers
2005 Oct 20
http://math.crg4.com/paper.pdf

James J. Tattersall
Elementary Number Theory in Nine Chapters , p99 , pp136-160
2e , 2005 , Cambridge Univ Press , ISBN 0521615240
correct definition of friendly

joseabdris , Gerry Myerson , Robert Israel
Distinct perfect squares with equal abundancy indices
2006 May 29
13 messages :

Jeff Probst and Bill Cross = Captain Nemo , et al.
Actuarial Outpost , Survivor II , Bonus Idol puzzle #20
2006 Jul 27 through 2006 Aug 29 , apparently
Messages #21 , #201 , #202 , #204 , #207 , etc.
Message #201 might not be what you generally expect from network tv

Richard F. Ryan
Results Concerning an Equation of Goormaghtigh and Related Topics
Int'l Mathematical Forum , 2006 , V1#25pp1195-1206

Keith Briggs
Abundant numbers and the Riemann hypothesis
Experimental Mathematics , 2006 , V15#2pp251-256
http://www.expmath.org/expmath/volumes/15/15.2/Briggs.pdf
computation on thinness of hypothetical violators of RH

Judy A. Holdener
Conditions Equivalent to the Existence of Odd Perfect Numbers
Mathematics Magazine , 2006 Dec , V79#5p389-391

Wm. G. Stanton , Judy A. Holdener
Abundancy "Outlaws" of the Form  (sigma(N) + t)/N
Journal of Integer Sequences , 2007 , V10 , Article 07.9.6
http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Holdener/holdener7.html

Laura Czarnecki , Judy A. Holdener
The Abundancy Index: Tracking Down Outlaws
2007
http://biology.kenyon.edu/HHMI/posters_2007/CzarneckiL.png

Jeffrey M. Ward
Does Ten Have a Friend?
2007
http://arxiv.org/pdf/0806.1001

Kevin A. Broughan , Qizhi Zhou
Odd multiperfect numbers of abundancy 4
University of Waikato, Hamilton, New Zealand
2007 May 3
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WKD-4NMWRJS-2&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=10&md5=1c31c4a5940660c4d8f279fd0b631d53
http://www.math.waikato.ac.nz/~kab/papers/perfect4o.pdf

Judy A. Holdener
Abundancy
painting
http://personal.kenyon.edu/holdenerj/Art/HoldenerPaintings.htm
http://personal.kenyon.edu/holdenerj/AbundancyDescription.htm

Walter Nissen
Exhaustive List of Primitive Friendly Pairs within a Rectangle
2007
http://upforthecount.com/math/ffp8.html

Amir Akbary , Zachary Friggstad , Robert Juricevic
Explicit upper bounds for $f(n)=\prod_{p_{\omega(n)}} \frac{p}{p-1}$
Contributions to Discrete Mathematics , 2007 , V2#2pp153-160
http://cdm.math.ucalgary.ca/index.php/cdm/article/view/77/50
C1.03 bound on abund ( n )

Oliver Knill
The oldest open problem in mathematics
2007 Dec 2
http://www.math.harvard.edu/~knill/seminars/perfect/handout.pdf
refers to abundancy as index , superabundant as index champion
illustrated by graphs of abundancy
gives Moral: "We only have to understand one function ..."
an attractive paper , still at a very early stage of perfection
a b                        a b
on p. 10 , for  p p  ( 3 instances ) read  p q
in Problem 14.2 , for  sigma ( n )  read  n

Walter Nissen
Augmentation of table of abundancies and non-abundancies
2008
http://upforthecount.com/math/black.html

Michel Marcus
Deficient primitive friendly pairs
2008 May 23
Extreme dpfp
2008 May 24
Unique dpfp ?
2008 May 24
Nice dpfps
2008 May 24

Michel Marcus
DPFP
2008 Jul
http://www.primepuzzles.net/puzzles/puzz_453.htm

Ed Pegg Jr.
http://www.mathpuzzle.com/
2008 Jul 27
See : "Abundant numbers"

J. Cislo , Marek Wolf
Criteria equivalent to the Riemann Hypothesis
2008 Aug 13 , arXiv:0808.0640v2
http://arxiv.org/abs/0808.0640
graph of sigma ( n )
re-iterates RH is , at least , very nearly true

Ziyue Guo , Judy A. Holdener
Abundancy Spiral -- Exploring Diagonal Patterns
2008
http://biology.kenyon.edu/HHMI/posters_2008/GuoZ.png

Kaitlin Rafferty , Judy A. Holdener
The Form of Perfect and Multiperfect Numbers
2008
http://biology.kenyon.edu/HHMI/posters_2008/RaffertyK.png

Richard F. Ryan
Improvements on Previous Abundancy Results
Int'l J. of Contemporary Mathematical Sciences , 2009 , V4#27p1299-1313
http://www.m-hikari.com/forth2/ryanIJCMS25-28-2009.pdf

Superabundant Numbers and the Riemann Hypothesis
Amer. Math. Monthly , 2009 Mar , V116#3p273-275
http://www.cs.uleth.ca/~akbary/07-0300.pdf

Daniel Baczkowski
Research Statement
fetched 2009 Aug
http://www.math.sc.edu/~baz/cv/research.pdf

G'erard P. Michon = Gerard P. Michon , Michel Marcus
Hemiperfect Numbers of Abundancy  sigma-sub-1 (n) = 11/2
Hemiperfect Numbers of Abundancy  sigma  (n) = 11/2
-1
2009 Jun 21
http://www.numericana.com/data/hpn11.htm

G'erard P. Michon = Gerard P. Michon , Michel Marcus
How Many Numbers of Abundancy  sigma-sub-1 (n) = 13/2 ?
How Many Numbers of Abundancy  sigma  (n) = 13/2 ?
-1
2009 Oct 26
http://www.numericana.com/data/hpn13.htm

G'erard P. Michon = Gerard P. Michon , Michel Marcus
Known Integers  n  of  Abundancy  sigma-sub-1 (n) = 15/2
Known Integers  n  of  Abundancy  sigma  (n) = 15/2
-1
2009 Nov 08
http://www.numericana.com/data/hpn15.htm

Karlik Zsuzsanna
A t”k‚letes sz mok = A tokeletes szamok = The perfect numbers
2009
http://www.cs.elte.hu/blobs/diplomamunkak/mattan/2009/karlik_zsuzsanna.pdf
thesis

GIMPS , the Great Internet Mersenne Prime Search
the search for 2-primitive friendly naturals = perfect numbers
http://www.mersenne.org

Numbers which are both amicable and friendly.
On-line Encyclopedia of Integer Sequences
http://research.att.com/~njas/sequences/A140688

The six tables of "Primitive Friendly Integers and Exclusive Multiples"
are represented in
Neil J. A. Sloane's On-Line Encyclopedia of Integer Sequences :

Solitary because prime powers

Solitary because coprime to sigma , but not prime powers

Solitary , but not coprime to sigma

Primitive Friendly

Friendly , not known to be Primitive Friendly

but not : Unknown , not known to be friendly nor solitary

- - - - -

All papers on perfect naturals , multiperfect naturals , abundant
naturals , deficient naturals , friendly naturals and harmonic mean
are necessarily based on abundancy ,
and some cover abundancy explicitly and extensively .

The perfects have abundancy = 2 .
Let  sigma ( )  be the multiplicative sum-of-divisors function ,
with  sigma ( m * n )  =  sigma ( m ) * sigma ( n )  for
m coprime n .
The multiperfects have integer abundancy and are
the zeroes of  sigma ( n ) mod n .
Friendly naturals have the same abundancy .

Abundancy is defined as the ratio of the multiplicative sum-of-divisors
function to the natural itself.

_                _   sigma ( n )
abundancy ( n )  =  abund ( n )   =   -----------
n

E.g.,  abund ( 10 )  =
sigma ( 10 ) / 10  =  (1+2+5+10) / 10  =  1.8  =  9 / 5 .

sigma ( n )  is the sum                    of the divisors of  n .
abund ( n )  is the sum of the reciprocals of the divisors of  n .
Thus , the notation  sigma  ( n )  has been used for  abundancy ( n ) .
-1
_
abundance ( n )  =  sigma ( n ) - 2n   is not  abundancy ( n ) .
_
excess ( n )  =  sigma ( n ) - 2n   is not  abundancy ( n ) .

p + 1
abund ( p )  =  -----
p

a+1
a        ( p   - 1 )
abund ( p  )  =  ------------
a
p  ( p - 1 )

1  <  abund ( n )
1  <  abund ( p )  <=  3/2
abund ( n )  <  abund ( m * n )

a                a+1
abund ( p  )  <  abund ( p    )

p + 1               a         p
-----  <=  abund ( p  )  <  -----
p                         p - 1

b                a
p  <  q   <===>   abund ( q  )  <  abund ( p  )

m and n  are friends  <===>  abund ( m ) = abund ( n )
If  n  has no friend , then  n  is solitary .

_           _  #-of-divisors ( n )
The harmonic mean  =  H ( n )  =  ------------------- .
abund ( n )

numer
Let  abund ( n )  =  ----- , with  numer coprime denom ,
denom

i.e. , numer / denom  is a reduced fraction .

Then ,
denom | n
numer | sigma ( n )
sigma ( denom )  <=  numer

2
Abundancy averages  pi  / 6 .
The range of abundancy is unbounded and , also , the number of
abundancies is unbounded , i.e. , the number of different values taken
on by the abundancy function is unbounded .
For  n > 5040 , abund ( n )  <   C1.78 * log ( log ( n ) ) ,
e     e
where  C1.78  =  e ^ gamma  ~=  1.7810724179901979852  and
gamma ~= .5772156649015328606  is Euler's constant ,
if and only if the Riemann hypothesis is true .
For  n >  120 , abund ( n )  <=  C1.84 * log ( log ( n ) ) ,
e     e
abund ( 180 )
where  C1.84  =  ---------------------  ~= 1.8413933941852926268765
log  ( log  ( 180 ) )
e      e
~=  1.033867784142748211 * C1.78 .

The abundancy champions , or record holders , are called superabundant .
A natural  n  is defined as superabundant if ,
for all  m < n , abund ( m ) < abund ( n ) .

Let  denom coprime sigma ( denom ) .
Let  abund ( denom )  =  abund ( friend ) .
Then , denom * sigma ( friend )  =  friend * sigma ( denom ) .
No factor of  denom  dividing  sigma ( denom ) ,
we must have  denom | friend .
But  abund ( denom )  <  abund ( any * denom ) .
Thus , friend = denom .
Thus , denom coprime sigma ( denom )  ====>  denom is solitary .

a
Thus , p  is solitary ,
i.e. , all primes and all prime powers are solitary .

Trivial cases are ignored above .

abund ( 1! ) = 1
abund ( 2! ) = 1 + 1/2
abund ( 3! ) = 2
abund ( 4! ) = 2 + 1/2
abund ( 5! ) = 3

The term "friendly numbers" has occasionally been misused to
refer to amicable naturals .
For the relationship , see "isotopic" on p. 5 of
Mariano Garcia , Jan Munch Pedersen , Herman te Riele
Amicable Pairs, a Survey
http://www.cwi.nl/ftp/CWIreports/MAS/MAS-R0307.pdf
and for the distinction , see
http://research.att.com/~njas/sequences/A140688
%N A140688 Numbers which are both amicable and friendly.
Perfects are those self-amicable .

GIMPS , the Great Internet Mersenne Prime Search , is a most advanced
part of the search for primitive friendly naturals , specifically
2-primitive friendly naturals .
http://www.mersenne.org

What are the most wanted abudancies ?
Perhaps :
very near 1 , especially < 384/361 , for friends
2 , especially an odd
12
13
14
15
17 / 2
19 / 2
very near 2
for odd :
5 / 3                     this would yield an odd perfect = 2-perfect
and a new 3-perfect
any integer                  this would be an odd multi-perfect
9 / 5                     this would be a friend of 10
12 / 7                     these would yield a new 3-perfect
8 / 5  ,  > 15              "
48 / 31                      "
32 / 21  ,  > 21             "
192 / 127                     "
128 / 85                      "
768 / 511  ,  > 1796165       "
512 / 341  ,  > 1023          "
3072 / 2047                    "
2048 / 1365                    "

Warm thanks are extended to Michel Marcus for his contributions .

2010 Mathematics Subject Classification : Primary 11-00 ; Secondary 01-00 , 01A60 , 01A61 , 01A70 , 11-03 , 11A05 , 11A25 , 11A41 , 11A51 , 11D61 , 11N25 , 11N60 , 11Y55

Walter Nissen

originally posted 2008-08-28
updated 2010-11-12