Main Page | See live article | Alphabetical index 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, ... (Sloane's A034897), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (Sloane's A034898). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (Sloane's A007592). The following table lists the first few k-hyperperfect numbers for some values of k, together with the Sloane number of the sequence of k-hyperperfect numbers: k Sequence First 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, ... 10   159841, ... 11   10693, ... 12 A028500 697, 2041, 1570153, 62722153, ... 18 A028501 1333, 1909, 2469601, 893748277, ... 30   3901, ... 31   214273, ... 35   306181, ... 48   26977, ... 60   24601, ... 66   296341, ... 78   486877, ... 108   275833, ... 132   96361, 130153, 495529,... 168   250321, ... 192   163201, ... 252   389593, ... 342   542413, ... 366   808861, ... 2772 A028502 95295817, 124035913, ... 31752 A034916 4660241041, 7220722321, ... It can be shown that if k > 1 is an odd integer and p = (3k + 1) / 2 and q = 3k + 4 are prime numbers, then p²q is k-hyperperfect; J. 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, and also that if k > and p = k + 1 is prime, then for all i > 1 such that q = pi - p + 1 is prime, pi - 1q is k-hyperperfect. Table of contents 1 Also see 2 External links 3 Further reading 3.1 Articles 3.2 Books Also see Multiply perfect number Deficient number Almost perfect number External links MathWorld: Hyperperfect number Further reading Articles Daniel Minoli, Robert Bear, Hyperperfect Numbers, PME Journal, 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. Books Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0071406158 (p.114-134) This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.