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Hyperperfect number

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In number theory, a k-hyperperfect number is a natural number n for which the equality holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.[1]

The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... (sequence A034897 in the OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in the OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in the OEIS).

List of hyperperfect numbers

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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:

List of some known k-hyperperfect numbers
k k-hyperperfect numbers OEIS
1 6, 28, 496, 8128, 33550336, ... OEISA000396
2 21, 2133, 19521, 176661, 129127041, ... OEISA007593
3 325, ...  
4 1950625, 1220640625, ...  
6 301, 16513, 60110701, 1977225901, ... OEISA028499
10 159841, ...  
11 10693, ...  
12 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... OEISA028500
18 1333, 1909, 2469601, 893748277, ... OEISA028501
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 95295817, 124035913, ... OEISA028502
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 4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ... OEISA034916
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 and are prime numbers, then is k-hyperperfect; Judson S. McCranie 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 pq are odd primes and k is an integer such that then pq is k-hyperperfect.

It is also possible to show that if k > 0 and is prime, then for all i > 1 such that is prime, is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:

Values of i for which n is k-hyperperfect
k Values of i OEIS
16 11, 21, 127, 149, 469, ... OEISA034922
22 17, 61, 445, ...  
28 33, 89, 101, ...  
36 67, 95, 341, ...  
42 4, 6, 42, 64, 65, ... OEISA034923
46 5, 11, 13, 53, 115, ... OEISA034924
52 21, 173, ...  
58 11, 117, ...  
72 21, 49, ...  
88 9, 41, 51, 109, 483, ... OEISA034925
96 6, 11, 34, ...  
100 3, 7, 9, 19, 29, 99, 145, ... OEISA034926

There are some even numbers which are hyperperfect for odd factors i.e., k * (sum of odd factors except 1 and itself) + 1 = number. e.g., the first 5 ones include 1300, 271872, 304640, 953344 and 1027584 for k = 3, 349, 353, 837 and 353. All odd hyperperfect numbers are odd factor hyperperfect numbers as they only have odd factors and have no even factors.

1300 has factors = 1, 2, 4, 5, 10, 13, 20, 25, 26, 50, 52, 65, 100, 130, 260, 325, 650, 1300

It has odd factors except 1 and itself = 5, 13, 25, 65, 325

Sum of odd factors except 1 and itself = 5 + 13 + 25 + 65 + 325 = 433

1300 - 1 = 1299 and 1299/433 = 3, an integer [citation needed] [clarification needed]

Hyperdeficiency

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The newly introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.

Definition (Minoli 2010): For any integer n and for integer k > 0, define the k-hyperdeficiency (or simply the hyperdeficiency) for the number n as

A number n is said to be k-hyperdeficient if

Note that for k = 1 one gets which is the standard traditional definition of deficiency.

Lemma: A number n is k-hyperperfect (including k = 1) if and only if the k-hyperdeficiency of n,

Lemma: A number n is k-hyperperfect (including k = 1) if and only if for some k, for at least one j > 0.

References

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  1. ^ Weisstein, Eric W. "Hyperperfect Number". mathworld.wolfram.com. Retrieved 2020-08-10.

Further reading

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Articles

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Books

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