Search a number
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  $k>1$  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 28 301 325 496 + 473651000101 840465000181 alternating 21 301 325 496 96361 + 147816121 569450701 amenable 21 28 301 325 496 + 972261181 974380921 apocalyptic 1333 2133 3901 8128 10693 + 24601 26977 arithmetic 21 301 697 1333 1909 + 9427657 9699181 binomial 21 28 325 496 8128 + 8589869056 137438691328 brilliant 21 697 1333 1909 3901 + 969890533 974380921 c.heptagonal 833857593493 c.nonagonal 28 325 496 8128 33550336 8589869056 137438691328 c.pentagonal 3901 163201 c.square 96361 cake 697 canyon 301 325 8128 congruent 21 28 301 325 496 + 8766061 9699181 Cunningham 28 325 2924101 2082096901 Curzon 21 19521 cyclic 697 1333 1909 3901 24601 + 9427657 9699181 D-number 21 d-powerful 2133 10693 214273 296341 de Polignac 1232053 1570153 1618597 2708413 4013833 + 61442077 61599553 deficient 21 301 325 697 1333 + 9427657 9699181 dig.balanced 21 10693 163201 1005649 2924101 + 181132801 194283181 Duffinian 21 301 325 697 1333 + 9427657 9699181 economical 21 301 1333 1909 2133 + 16904101 18116737 emirpimes 2041 389593 1055833 1232053 1284121 + 97438381 97585249 equidigital 21 301 1333 1909 2133 + 16904101 18116737 esthetic 21 evil 325 697 1333 1909 176661 + 939137257 941755189 fibodiv 28 Fibonacci 21 frugal 176661 214273 1433701 1950625 10445221 + 822000961 925572421 gapful 159841 1570153 happy 28 301 496 1333 2133 + 1284121 6392257 harmonic 28 496 8128 33550336 8589869056 137438691328 Harshad 21 2133 176661 129127041 8589869056 heptagonal 697 hexagonal 28 325 496 8128 33550336 8589869056 137438691328 hoax 4013833 8883841 14489437 30374941 36640993 + 46024681 79001833 Hogben 21 1333 16513 37021141 65618101 + 1453482210421 1951956868501 iban 21 301 2041 214273 idoneal 21 28 inconsummate 26977 interprime 21 1055833 1787917 12283693 38749153 Jacobsthal 21 junction 19521 495529 1055833 1950625 7478041 + 79001833 94472041 katadrome 21 Lehmer 250321 lucky 21 2133 19521 163201 214273 + 3328921 3420301 magnanimous 21 metadrome 28 modest 1333 2133 Moran 21 Motzkin 21 mountain 496 697 19521 159841 nialpdrome 21 nonagonal 325 nude 8128 oban 28 325 697 octagonal 21 2133 19521 176661 129127041 odious 21 28 301 496 2041 + 972261181 974380921 pancake 301 25898945437 panconsummate 21 pandigital 21 perfect 28 496 8128 33550336 8589869056 137438691328 pernicious 21 28 301 496 2133 + 9398593 9699181 plaindrome 28 1333 practical 28 496 8128 productive 28 Proth 112803841 463743221761 pseudoperfect 28 496 8128 33550336 repfigit 28 repunit 21 1333 16513 37021141 65618101 + 1453482210421 1951956868501 self 10693 214273 1787917 2469601 2768581 + 924642337 972261181 semiprime 21 301 697 1333 1909 + 97438381 97585249 Smith 4013833 8883841 14489437 30374941 36640993 + 46024681 79001833 sphenic 1570153 star 3901 super-d 10693 306181 808861 1005649 1055833 + 7152001 9699181 triangular 21 28 325 496 8128 + 8589869056 137438691328 uban 21 28 Ulam 28 8128 51301 163201 486877 + 1570153 4013833 unprimeable 325 upside-down 28 wasteful 28 325 496 697 2041 + 9427657 9699181 Zumkeller 28 496 8128