- Source: Hyperperfect number
In number theory, a k-hyperperfect number is a natural number n for which the equality
n
=
1
+
k
(
σ
(
n
)
−
n
−
1
)
{\displaystyle n=1+k(\sigma (n)-n-1)}
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.
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
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:
It can be shown that if k > 1 is an odd integer and
p
=
3
k
+
1
2
{\displaystyle p={\tfrac {3k+1}{2}}}
and
q
=
3
k
+
4
{\displaystyle q=3k+4}
are prime numbers, then
p
2
q
{\displaystyle p^{2}q}
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
)
=
p
q
−
1
,
{\displaystyle k(p+q)=pq-1,}
then pq is k-hyperperfect.
It is also possible to show that if k > 0 and
p
=
k
+
1
{\displaystyle p=k+1}
is prime, then for all i > 1 such that
q
=
p
i
−
p
+
1
{\displaystyle q=p^{i}-p+1}
is prime,
n
=
p
i
−
1
q
{\displaystyle 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:
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 > 0, define the k-hyperdeficiency (or simply the hyperdeficiency) for the number n as
δ
k
(
n
)
=
n
(
k
+
1
)
+
(
k
−
1
)
−
k
σ
(
n
)
{\displaystyle \delta _{k}(n)=n(k+1)+(k-1)-k\sigma (n)}
A number n is said to be k-hyperdeficient if
δ
k
(
n
)
>
0.
{\displaystyle \delta _{k}(n)>0.}
Note that for k = 1 one gets
δ
1
(
n
)
=
2
n
−
σ
(
n
)
,
{\displaystyle \delta _{1}(n)=2n-\sigma (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,
δ
k
(
n
)
=
0.
{\displaystyle \delta _{k}(n)=0.}
Lemma: A number n is k-hyperperfect (including k = 1) if and only if for some k,
δ
k
−
j
(
n
)
=
−
δ
k
+
j
(
n
)
{\displaystyle \delta {k-j}(n)=-\delta _{k+j}(n)}
for at least one j > 0.
References
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 114. ISBN 1-4020-4215-9. Zbl 1151.11300.
Further reading
= Articles
=Minoli, Daniel; Bear, Robert (Fall 1975), "Hyperperfect numbers", Pi Mu Epsilon Journal, 6 (3): 153–157.
Minoli, Daniel (Dec 1978), "Sufficient forms for generalized perfect numbers", Annales de la Faculté des Sciences UNAZA, 4 (2): 277–302.
Minoli, Daniel (Feb 1981), "Structural issues for hyperperfect numbers", Fibonacci Quarterly, 19 (1): 6–14, doi:10.1080/00150517.1981.12430116.
Minoli, Daniel (April 1980), "Issues in non-linear hyperperfect numbers", Mathematics of Computation, 34 (150): 639–645, doi:10.2307/2006107, JSTOR 2006107.
Minoli, Daniel (October 1980), "New results for hyperperfect numbers", Abstracts of the American Mathematical Society, 1 (6): 561.
Minoli, Daniel; Nakamine, W. (1980). "Mersenne numbers rooted on 3 for number theoretic transforms". ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 5. pp. 243–247. doi:10.1109/ICASSP.1980.1170906..
McCranie, Judson S. (2000), "A study of hyperperfect numbers", Journal of Integer Sequences, 3: 13, Bibcode:2000JIntS...3...13M, archived from the original on 2004-04-05.
te Riele, Herman J.J. (1981), "Hyperperfect numbers with three different prime factors", Math. Comp., 36 (153): 297–298, doi:10.1090/s0025-5718-1981-0595066-9, MR 0595066, Zbl 0452.10005.
te Riele, Herman J.J. (1984), "Rules for constructing hyperperfect numbers", Fibonacci Q., 22: 50–60, doi:10.1080/00150517.1984.12429920, Zbl 0531.10005.
= Books
=Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p. 114-134)
External links
MathWorld: Hyperperfect number
A long list of hyperperfect numbers under Data
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