HYPERPERFECT NUMBER

In mathematics, a '''k''-hyperperfect number' (sometimes just called ''hyperperfect number'') is a natural number ''n'' for which the equality ''n'' = 1 + ''k''(''σ''(''n'') − ''n'' − 1) holds, where ''σ''(''n'') is the divisor function (i.e., the sum of all positive divisors of ''n''). A number is perfect iff it is 1-hyperperfect.
The first few numbers in the sequence of ''k''-hyperperfect numbers are 6, 21, 28, 301, 325, 496, ... (sequence A034897 in OEIS), with the corresponding values of ''k'' being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in OEIS). The first few ''k''-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in OEIS).

Contents

List of hyperperfect numbers

External links

Further reading

Articles

Books

List of hyperperfect numbers

The following table lists the first few ''k''-hyperperfect numbers for some values of ''k'', together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of ''k''-hyperperfect numbers:

''k''	OEIS	Some known ''k''-hyperperfect numbers
1	A000396	6, 28, 496, 8128, 33550336, ...
2	A007593	21, 2133, 19521, 176661, 129127041, ...
3		325, ...
4		1950625, 1220640625, ...
6	A028499	301, 16513, 60110701, 1977225901, ...
10		159841, ...
11		10693, ...
12	A028500	697, 2041, 1570153, 62722153, 10604156641, 13544168521, ...
18	A028501	1333, 1909, 2469601, 893748277, ...
19		51301, ...
30		3901, 28600321, ...
31		214273, ...
35		306181, ...
40		115788961, ...
48		26977, 9560844577, ...
59		1433701, ...
60		24601, ...
66		296341, ...
75		2924101, ...
78		486877, ...
91		5199013, ...
100		10509080401, ...
108		275833, ...
126		12161963773, ...
132		96361, 130153, 495529, ...
136		156276648817, ...
138		46727970517, 51886178401, ...
140		1118457481, ...
168		250321, ...
174		7744461466717, ...
180		12211188308281, ...
190		1167773821, ...
192		163201, 137008036993, ...
198		1564317613, ...
206		626946794653, 54114833564509, ...
222		348231627849277, ...
228		391854937, 102744892633, 3710434289467, ...
252		389593, 1218260233, ...
276		72315968283289, ...
282		8898807853477, ...
296		444574821937, ...
342		542413, 26199602893, ...
348		66239465233897, ...
350		140460782701, ...
360		23911458481, ...
366		808861, ...
372		2469439417, ...
396		8432772615433, ...
402		8942902453, 813535908179653, ...
408		1238906223697, ...
414		8062678298557, ...
430		124528653669661, ...
438		6287557453, ...
480		1324790832961, ...
522		723378252872773, 106049331638192773, ...
546		211125067071829, ...
570		1345711391461, 5810517340434661, ...
660		13786783637881, ...
672		142718568339485377, ...
684		154643791177, ...
774		8695993590900027, ...
810		5646270598021, ...
814		31571188513, ...
816		31571188513, ...
820		1119337766869561, ...
968		52335185632753, ...
972		289085338292617, ...
978		60246544949557, ...
1050		64169172901, ...
1410		80293806421, ...
2772	A028502	95295817, 124035913, ...
3918		61442077, 217033693, 12059549149, 60174845917, ...
9222		404458477, 3426618541, 8983131757, 13027827181, ...
9828		432373033, 2797540201, 3777981481, 13197765673, ...
14280		848374801, 2324355601, 4390957201, 16498569361, ...
23730		2288948341, 3102982261, 6861054901, 30897836341, ...
31752	A034916	4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ...
55848		15166641361, 44783952721, 67623550801, ...
67782		18407557741, 18444431149, 34939858669, ...
92568		50611924273, 64781493169, 84213367729, ...
100932		50969246953, 53192980777, 82145123113, ...

It can be shown that if ''k'' > 1 is an odd integer and ''p'' = (3''k'' + 1) / 2 and ''q'' = 3''k'' + 4 are prime numbers, then ''p''²''q'' is ''k''-hyperperfect; Judson S. McCraine has conjectured in 2000 that all ''k''-hyperperfect numbers for odd ''k'' > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if ''p'' ≠ ''q'' are odd primes and ''k'' is an integer such that ''k''(''p'' + ''q'') = ''pq'' - 1, then ''pq'' is ''k''-hyperperfect.
It is also possible to show that if ''k'' > 0 and ''p'' = ''k'' + 1 is prime, then for all ''i'' > 1 such that ''q'' = ''p''^''i'' − ''p'' + 1 is prime, ''n'' = ''p''^{''i'' − 1}''q'' is ''k''-hyperperfect. The following table lists known values of ''k'' and corresponding values of ''i'' for which ''n'' is ''k''-hyperperfect:

''k''	OEIS	Values of ''i''
16	A034922	11, 21, 127, 149, 469, ...
22		17, 61, 445, ...
28		33, 89, 101, ...
36		67, 95, 341, ...
42	A034923	4, 6, 42, 64, 65, ...
46	A034924	5, 11, 13, 53, 115, ...
52		21, 173, ...
58		11, 117, ...
72		21, 49, ...
88	A034925	9, 41, 51, 109, 483, ...
96		6, 11, 34, ...
100	A034926	3, 7, 9, 19, 29, 99, 145, ...

External links

★ MathWorld: Hyperperfect number

Further reading

Articles

★ Daniel Minoli, Robert Bear, ''Hyperperfect Numbers'', PME (Pi Mu Epsilon) Journal, University Oklahoma, Fall 1975, pp. 153-157.

★ Daniel Minoli, ''Sufficient Forms For Generalized Perfect Numbers'', Ann. Fac. Sciences, Univ. Nation. Zaire, Section Mathem; Vol. 4, No. 2, Dec 1978, pp. 277-302.

★ Daniel Minoli, ''Structural Issues For Hyperperfect Numbers'', Fibonacci Quarterly, Feb. 1981, Vol. 19, No. 1, pp. 6-14.

★ Daniel Minoli, ''Issues In Non-Linear Hyperperfect Numbers, Mathematics of Computation'', Vol. 34, No. 150, April 1980, pp. 639-645.

★ Daniel Minoli, ''New Results For Hyperperfect Numbers'', Abstracts American Math. Soc., October 1980, Issue 6, Vol. 1, pp. 561.

★ Daniel Minoli, W. Nakamine, ''Mersenne Numbers Rooted On 3 For Number Theoretic Transforms'', 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.

★ Judson S. McCranie, ''A Study of Hyperperfect Numbers'', Journal of Integer Sequences, Vol. 3 (2000), http://www.math.uwaterloo.ca/JIS/VOL3/mccranie.html

Books

★ Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)