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hyperperfect numbers
A number  $n$  is said to be  $k$-hyperperfect if  $n=1+k(\sigma(n)-n-1)$.

For example, 301 is 6-hyperperfect since  $301=1+6\cdot(\sigma(301)-301-1)$.

In general, a hyperperfect number is a number which is  $k$-hyperperfect for some integer  $k$.

Perfect numbers are 1-hyperperfect, since  $n=1+\sigma(n)-n-1$  is equivalent to the condition  $\sigma(n)=2n$.

Jud McCranie has conjectured that all  $k$-hyperperfect numbers for  $k1$">  odd are of the form  $p^2\cdot q$, where  $p=(3k + 1) / 2$  and  $q = 3k + 4$  are two prime numbers. For example, this happens for  $k$  equal to 3, 11, 19, 31, 35, 59, 75, 91, 111,...

The first hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, 1333, 1909, 2041, 2133, 3901, 8128, 10693, 16513, 19521, 24601 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of hyperperfect numbers

A graph displaying how many hyperperfect numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of hyperperfect numbers
Useful links Mathworld, Hyperperfect Number
Wikipedia, Hyperperfect number
OEIS, Sequence A034897
Hyperperfect numbers can also be... (you may click on names or numbers and on + to get more values)

aban 21 28 + 256929000301 300609000733 473651000101 840465000181 alternating 21 301 + 65618101 125438101 147816121 569450701 amenable 21 28 + 941755189 969890533 972261181 974380921 apocalyptic 1333 2133 + 16513 19521 24601 26977 arithmetic 21 301 + 8766061 8883841 9427657 9699181 binomial 21 28 + 8128 33550336 8589869056 137438691328 brilliant 21 697 + 914054329 922604701 969890533 974380921 c.heptagonal 833857593493 c.nonagonal 28 325 + 8128 33550336 8589869056 137438691328 c.pentagonal 3901 163201 c.square 96361 cake 697 congruent 21 28 + 4304341 5199013 8766061 9699181 Cunningham 28 325 2924101 2082096901 Curzon 21 19521 cyclic 697 1333 + 8766061 8883841 9427657 9699181 D-number 21 d-powerful 2133 10693 214273 296341 de Polignac 1232053 1570153 + 47842237 55262737 61442077 61599553 deficient 21 301 + 8883841 9398593 9427657 9699181 dig.balanced 21 10693 + 65618101 138786229 181132801 194283181 Duffinian 21 301 + 8883841 9398593 9427657 9699181 economical 21 301 + 16260901 16641241 16904101 18116737 emirpimes 2041 389593 + 94472041 95295817 97438381 97585249 equidigital 21 301 + 16260901 16641241 16904101 18116737 esthetic 21 evil 325 697 + 922604701 925572421 939137257 941755189 fibodiv 28 Fibonacci 21 frugal 176661 214273 + 542935873 573685201 822000961 925572421 gapful 159841 1570153 happy 28 301 + 159841 1055833 1284121 6392257 harmonic 28 496 8128 33550336 8589869056 137438691328 Harshad 21 2133 176661 129127041 8589869056 heptagonal 697 hexagonal 28 325 + 8128 33550336 8589869056 137438691328 hoax 4013833 8883841 + 36640993 38749153 46024681 79001833 Hogben 21 1333 + 1008025036021 1421971583761 1453482210421 1951956868501 iban 21 301 2041 214273 idoneal 21 28 inconsummate 26977 interprime 21 1055833 1787917 12283693 38749153 Jacobsthal 21 junction 19521 495529 + 55262737 62804941 79001833 94472041 katadrome 21 Lehmer 250321 lucky 21 2133 + 1063141 2488201 3328921 3420301 magnanimous 21 metadrome 28 modest 1333 2133 Moran 21 Motzkin 21 nialpdrome 21 nonagonal 325 nude 8128 oban 28 325 697 octagonal 21 2133 19521 176661 129127041 odious 21 28 + 940209541 969890533 972261181 974380921 pancake 301 25898945437 panconsummate 21 pandigital 21 perfect 28 496 8128 33550336 8589869056 137438691328 pernicious 21 28 + 7478041 8766061 9398593 9699181 plaindrome 28 1333 practical 28 496 8128 Proth 112803841 463743221761 pseudoperfect 28 496 8128 33550336 repfigit 28 repunit 21 1333 + 1008025036021 1421971583761 1453482210421 1951956868501 self 10693 214273 + 828884629 836729233 924642337 972261181 semiprime 21 301 + 95235601 95295817 97438381 97585249 Smith 4013833 8883841 + 36640993 38749153 46024681 79001833 sphenic 1570153 star 3901 super-d 10693 306181 + 4312681 4658449 7152001 9699181 triangular 21 28 + 8128 33550336 8589869056 137438691328 uban 21 28 Ulam 28 8128 + 1005649 1433701 1570153 4013833 unprimeable 325 upside-down 28 wasteful 28 325 + 8766061 8883841 9427657 9699181 Zumkeller 28 496 8128