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 |
|
2 External links 3 Further reading |
Also see
External links
Further reading
Articles
Books