( 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 .Warm thanks are extended to Michel Marcus for his contributions .A little about 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 Academiae Scientiarium Petropolitanae http://books.google.com/books?id=iOUGAAAAYAAJ&pg=PA88 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/ http://books.google.com/books?id=mbk9AAAAYAAJ free online full text 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 www.copyright.gov/circs/circ22.pdf 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 Fr`eres , 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://links.jstor.org/sici?sici=0002.9947(194411)56:3<448:OHCASN>2.0.CO;2.S 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 http://www.bestwebbuys.com/9780486656205 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 Advanced Problem 6020* 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 Advanced Problem 6070 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 hypoth`ese 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://links.jstor.org/sici?sici=0025-5718%28199110%2957%3A196%3C857%3AITFLBF%3E2.0.CO%3B2-K http://www.jstor.org/stable/2938723 Walter Nissen Exponential Prime Power Representation 1995 May 23 http://groups.google.com/group/sci.math/msg/9d0e37530f0a756d http://upforthecount.com/math/epprep.html Kevin S. Brown The Distribution of Perfection 1995 May 27 http://groups.google.com/group/sci.math/msg/9db3d3a76ad7423b http://www.mathpages.com/home/kmath223.htm Chris E. Thompson re: The Distribution of Perfection 1995 May 29 http://groups.google.com/group/sci.math/msg/7b41bc07772a6b6b 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 http://groups.google.com/group/sci.math/msg/288219f421f27d41 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 http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ55787.pdf 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://links.jstor.org/sici?sici=0025-570X(200310)76:4<299:ASDPRT>2.0.CO;2-O 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 http://www.bestwebbuys.com/9780387208602 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 see comments thereon at http://math.crg4.com/ James J. Tattersall Elementary Number Theory in Nine Chapters , p99 , pp136-160 2e , 2005 , Cambridge Univ Press , ISBN 0521615240 http://www.bestwebbuys.com/9780521615242 correct definition of friendly joseabdris , Gerry Myerson , Robert Israel Distinct perfect squares with equal abundancy indices 2006 May 29 http://groups.google.com/group/sci.math/msg/c9901d8a2e03e8f4 13 messages : http://groups.google.com/group/sci.math/browse_thread/thread/a68aa1190a8dbbd7/c9901d8a2e03e8f4#c9901d8a2e03e8f4 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 http://actuary.ca/actuarial_discussion_forum/showthread.php?t=87324&page=3 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 http://www.m-hikari.com/imf-password/25-28-2006/ryanIMF25-28-2006.pdf 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 http://tech.groups.yahoo.com/group/primenumbers/message/19376?var=0 Extreme dpfp 2008 May 24 http://tech.groups.yahoo.com/group/primenumbers/message/19379?var=0 Unique dpfp ? 2008 May 24 http://tech.groups.yahoo.com/group/primenumbers/message/19380?var=0 Nice dpfps 2008 May 24 http://tech.groups.yahoo.com/group/primenumbers/message/19381?var=0 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 Amir Akbary , Zachary Friggstad 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 - - - - -A little about abundancy 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 "

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