View source for Abundant number

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300701.png" /> denote the sum of the distinct divisors of an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300702.png" /> (cf. [[Divisor|Divisor]]; [[Number of divisors|Number of divisors]]). The integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300703.png" /> is called abundant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300704.png" />; deficient if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300705.png" />; and perfect if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300706.png" /> (cf. also [[Perfect number|Perfect number]]). Note that some authors call a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300707.png" /> abundant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300708.png" />. Clearly, these numbers are in fact perfect or abundant (i.e.  "non-deficient" ) numbers.

In [[#References|[a5]]], L.E. Dickson gives details on the early history of abundant numbers. G. Nicomachus (about 100) separated the even numbers into abundant, deficient and perfect, and dwelled on the ethical importance of the three types. A.M.S. Boethius (around 500), in a Latin exposition of the arithmetic of Nicomachus, stated that perfect numbers are rare, while abundant ( "superfluous" ) and deficient ( "diminutos" ) numbers are found to an unlimited extent. N. Jordanus (around 1236) stated that every multiple of a perfect or abundant number is abundant. He attempted to prove the erroneous statement that all abundant numbers are even. C. Bovillus (around 1509) corrected this statement, by citing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a1300709.png" /> and its multiples. Bachet de Méziriac (around 1600) gave a proof that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007010.png" /> is perfect if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007011.png" /> is a [[Prime number|prime number]], and abundant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007012.png" /> is composite. He remarked that the odd number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007013.png" /> is abundant. J. Broscius (around 1652) showed that there are only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007014.png" /> abundant numbers between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007016.png" /> and all of them are even; the only odd abundant number less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007017.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007018.png" />. (The statement by E. Lucas (1891) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007019.png" /> is the smallest odd abundant number is probably a misprint for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007020.png" />.) Ch. de Neuveglise (1700) proved that the products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007021.png" /> of two consecutive numbers are abundant, and all multiplies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007022.png" /> or an abundant number are abundant. J. Struve (1808) considered abundant numbers which are products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007023.png" /> of three distinct prime numbers in ascending order; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007026.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007027.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007031.png" /> is abundant for any prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007032.png" />. Of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007034.png" /> are abundant.

Dickson (1913, [[#References|[a6]]]) called a non-deficient number primitive abundant if it is not a multiple of a smaller non-deficient number. He proved that there are only a finite number of primitive non-deficient numbers having a given number of distinct odd prime factors and a given number of factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007035.png" />.

There is no odd abundant number with fewer than three distinct prime factors, the primitive ones with three are

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He gave also a table of all even abundant numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007038.png" />. Dickson's result was a starting point for much further research. In 1949 and 1968, H.N. Shapiro ([[#References|[a20]]], [[#References|[a21]]]) proved the following result. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007039.png" /> be a rational number. A necessary and sufficient condition that there exist infinitely many primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007041.png" />-abundant numbers (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007042.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007043.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007045.png" />) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007046.png" /> distinct prime factors is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007047.png" /> has a representation

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with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007051.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007052.png" /> is the Euler [[Totient function|totient function]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007053.png" /> denotes the number of distinct prime factors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007054.png" />.

In 1933, F. Behrend, H. Davenport and S. Chowla [[#References|[a4]]] showed that the density of non-deficient numbers exists and is positive. This result follows also from a theorem of P. Erdős [[#References|[a7]]] stating that the sum of reciprocals of primitive abundant numbers converges. Let

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be the counting function of primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007056.png" />-abundant numbers. Erdős proved that [[#References|[a10]]]

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and that [[#References|[a8]]]

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This was sharpened successively by A. Ivić [[#References|[a13]]], with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007060.png" /> in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007062.png" /> in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007063.png" />; and by M.R. Avidon [[#References|[a2]]], who considered <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007064.png" /> in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007065.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007066.png" /> in place of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007067.png" />.

L. Alaoglu and Erdős [[#References|[a1]]] call a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007068.png" /> superabundant if

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007070.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007071.png" /> be the counting function of superabundant numbers. For two consecutive superabundant numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007073.png" /> they prove that

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and this was sharpened to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007075.png" /> for an infinity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007076.png" /> by J.-L. Nicolas [[#References|[a16]]]. Alaoglu and Erdős showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007077.png" />, while Erdős and Nicolas [[#References|[a11]]] demonstrated that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007078.png" />. Alaoglu and Erdős [[#References|[a1]]] introduced also the notion of highly abundant number, a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007079.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007080.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007081.png" />. For the counting function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007082.png" /> of these numbers one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007083.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007084.png" /> and large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007085.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007086.png" /> is highly abundant, then the largest prime factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007087.png" /> is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007088.png" />.

Erdős and Nicolas [[#References|[a11]]] call a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007089.png" /> cube-free superabundant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007090.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007091.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007092.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007094.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007095.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007096.png" /> a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007097.png" /> a positive integer). They prove that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a13007099.png" /> are two consecutive cube-free superabundant numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070100.png" />. A non-deficient number is called weird by S.J. Benkovski and Erdős [[#References|[a3]]] if it is not pseudo-perfect (cf. also [[Perfect number|Perfect number]]). They proved that the density of weird numbers is positive.

V. Siva Rama Prasad and D.R. Reddy [[#References|[a23]]] say that a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070101.png" /> is primitive unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070103.png" />-abundant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070104.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070105.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070107.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070108.png" />). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070109.png" /> denotes the sum of unitary divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070110.png" /> (for these functions, as well as related results, see also [[#References|[a15]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070111.png" /> be the set of these numbers. Then

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==Miscellaneous results.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070113.png" />. A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070114.png" /> is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070116.png" />-non-deficient if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070117.png" />. By sharpening a result of O. Grün [[#References|[a12]]], H. Salié [[#References|[a18]]] proved that the least prime factor of every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070118.png" />-non-deficient number with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070119.png" /> prime factors is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070120.png" />.

Ch.R. Wall [[#References|[a24]]] proved that there exist infinitely many abundant integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070121.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070123.png" /> given). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070124.png" /> be fixed. Then there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070125.png" /> consecutive abundant numbers. There exist infinitely many sequences of five consecutive deficient numbers. (See [[#References|[a25]]].) See [[#References|[a14]]] for a table of odd primitive abundant numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070126.png" /> with five distinct prime factors for which

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070128.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070129.png" /> is abundant, see [[#References|[a22]]].

For others results on deficient, perfect, or related numbers, see [[#References|[a15]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a19]]], [[#References|[a17]]].

L. Moser [[#References|[a26]]] proved that every integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070130.png" /> can be expressed as the sum of two abundant numbers. Actually, this is valid for integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070131.png" />, see [[#References|[a27]]].

For a table of abundant numbers less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070132.png" />, see [[#References|[a28]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Alaoglu,   P. Erdös,   "On highly composite and similar numbers"  ''Trans. Amer. Math. Soc.'' , '''56'''  (1944)  pp. 448–469</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.R. Avidon,   "On the distribution of primitive abundant numbers"  ''Acta Arith.'' , '''77'''  (1996)  pp. 195–205</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.J. Benkovski,   P. Erdös,   "On weird and pseudoperfect numbers"  ''Math. Comput.'' , '''28'''  (1974)  pp. 617–623</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Davenport,   "Über numeri abundantes"  ''Preuss. Akad. Wiss. Sitzungsber'' , '''26/29'''  (1933)  pp. 830–837</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.E. Dickson,   "History of the theory of numbers" , '''I (Divisibility and primality)''' , Chelsea  (1919)  (Reprint: AMS 1999)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L.E. Dickson,   "Finiteness of odd perfect and primitive abundent numbers with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070133.png" /> distinct prime factors"  ''Amer. J. Math.'' , '''35'''  (1913)  pp. 413–422</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P. Erdös,   "On the density of the abundant numbers"  ''J. London Math. Soc.'' , '''9'''  (1934)  pp. 278–282</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P. Erdös,   "On primitive abundant numbers"  ''J. London Math. Soc.'' , '''9'''  (1935)  pp. 49–58</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P. Erdös,   "Note on consecutive abundant numbers"  ''J. London Math. Soc.'' , '''13'''  (1938)  pp. 128–131</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  P. Erdös,   "Remarks on number theory I, On primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070134.png" />-abundant numbers"  ''Acta Arith.'' , '''5'''  (1958)  pp. 25–33</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  P. Erdös,   J.-L. Nicolas,   "Répartition des nombres superabondantes"  ''Bull. Soc. Math. France'' , '''103'''  (1975)  pp. 65–90</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  O. Grün,   "Über ungerade vollkommene Zahlen"  ''Math. Z.'' , '''55'''  (1952)  pp. 353–354</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  A. Ivić,   "The distribution of primitive abundant numbers"  ''Studia Sci. Math. Hung.'' , '''20'''  (1985)  pp. 183–187</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  M. Kishore,   "Odd integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070135.png" /> with five distinct prime factors for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070136.png" />"  ''Math. Comput.'' , '''32'''  (1978)  pp. 303–309</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  D.S. Mitrinović,   J. Sándor,   "Handbook of number theory" , Kluwer Acad. Publ.  (1995)  (In coop. with B. Crstici)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  J.-L. Nicolas,   "Ordre maximal d'un élément du groupe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130070/a130070137.png" /> des permutations et `highly composite numbers'"  ''Bull. Soc. Math. France'' , '''97'''  (1969)  pp. 129–191</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  H.J.J. te Riele,   "A theoretical and computational study of generalized aliquot sequences"  ''Math. Centrum, Amsterdam''  (1975)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  H. Salié,   "Über abundante Zahlen"  ''Math. Nachr.'' , '''9'''  (1953)  pp. 217–220</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  J. Sándor,   "On a method of Galambos and Kátai concerning the frequency of deficient numbers"  ''Publ. Math. (Debrecen)'' , '''39'''  (1991)  pp. 155–157</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  H.N. Shapiro,   "Note on a theorem of Dickson"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 450–452</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  H.N. Shapiro,   "On primitive abundant numbers"  ''Commun. Pure Appl. Math.'' , '''21'''  (1968)  pp. 111–118</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  W. Sierpinski,   "Teoria liczb" , '''II''' , Warsawa  (1959)</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  V. Siva Rama Prasad,   D.R. Reddy,   "On primitive unitary abundant numbers"  ''Indian J. Pure Appl. Math.'' , '''21'''  (1990)  pp. 40–44</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  Ch.R. Wall,   "Problem E3002"  ''Amer. Math. Monthly'' , '''90'''  (1983)  pp. 400  (Solution by N.J. Fine: 93 (1986), 814)</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  Ch.R. Wall,   "Problem, 6356"  ''Amer. Math. Monthly'' , '''88'''  (1981)  pp. 623  (Solution by L.L. Foster: 90 (1983), 215-216)</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  L. Moser,   "Problem E848"  ''Amer. Math. Monthly'' , '''56'''  (1949)  pp. 478</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  E. Bach,   J. Shallit,   "Algorithmic number theory" , MIT  (1996)  pp. 334</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top">  J.W.L. Glaiser,   "Number-Divisor Tables" , British Assoc. Math. Tables  (1940)</TD></TR></table>