Hyperperfect number
Encyclopedia
In mathematics
, a k-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 divisor
s of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect number
s, which are 1-hyperperfect.
The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, ... , with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... . The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... .
(OEIS) of the sequence of k-hyperperfect numbers:
It can be shown that if k > 1 is an odd
integer
and p = (3k + 1) / 2 and q = 3k + 4 are prime number
s, then p²q 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 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 = pi − p + 1 is prime, n = pi − 1q is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:
Definition (Minoli 2010): For any integer n and for integer k, -∞
δk(n) = n(k+1) +(k-1) –kσ(n)
A number n is said to be k-hyperdeficient if δk(n) > 0.
Note that for k=1 one gets δ1(n)= 2n–σ(n), 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, δk(n) = 0.
Lemma: A number n is k-hyperperfect (including k=1) if and only if for some k, δk-j(n) = -δk+j(n) for at least one j > 0.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a k-hyperperfect number is a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
(i.e., the sum of all positive divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
s of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
s, which are 1-hyperperfect.
The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, ... , with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... . The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... .
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 SequencesOn-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
(OEIS) of the sequence of k-hyperperfect numbers:
| k | OEIS | Some known k-hyperperfect numbers |
|---|---|---|
| 1 | 6, 28, 496, 8128, 33550336, ... | |
| 2 | 21, 2133, 19521, 176661, 129127041, ... | |
| 3 | 325, ... | |
| 4 | 1950625, 1220640625, ... | |
| 6 | 301, 16513, 60110701, 1977225901, ... | |
| 10 | 159841, ... | |
| 11 | 10693, ... | |
| 12 | 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... | |
| 18 | 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 | 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 | 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
Even and odd numbers
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
and p = (3k + 1) / 2 and q = 3k + 4 are prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s, then p²q 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 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 = pi − p + 1 is prime, n = pi − 1q 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 | 11, 21, 127, 149, 469, ... | |
| 22 | 17, 61, 445, ... | |
| 28 | 33, 89, 101, ... | |
| 36 | 67, 95, 341, ... | |
| 42 | 4, 6, 42, 64, 65, ... | |
| 46 | 5, 11, 13, 53, 115, ... | |
| 52 | 21, 173, ... | |
| 58 | 11, 117, ... | |
| 72 | 21, 49, ... | |
| 88 | 9, 41, 51, 109, 483, ... | |
| 96 | 6, 11, 34, ... | |
| 100 | 3, 7, 9, 19, 29, 99, 145, ... |
Hyperdeficiency
The newly-introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.Definition (Minoli 2010): For any integer n and for integer k, -∞
δk(n) = n(k+1) +(k-1) –kσ(n)
A number n is said to be k-hyperdeficient if δk(n) > 0.
Note that for k=1 one gets δ1(n)= 2n–σ(n), 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, δk(n) = 0.
Lemma: A number n is k-hyperperfect (including k=1) if and only if for some k, δk-j(n) = -δk+j(n) for at least one j > 0.
Books
- Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)