Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.3 |

1680 Westfield Court

Lawrenceville, GA 30043

USA

Email address: jud.mccranie@mindspring.com

*Dedicated to the hyperperfect Anne McCranie, age 28 months.*

**Abstract:**
A number *n* is *k*-hyperperfect for some integer *k* if
*n* = 1 + *k s*(*n*),
where *s*(*n*) is the sum of the proper divisors of *n*.
The 1-hyperperfect numbers are the familiar perfect numbers.
This paper presents
some theorems, conjectures and tables concerning hyperperfect numbers.
All hyperperfect numbers less than 10^{11} have been computed.
Evidence is presented suggesting that a
published conjecture is false.

*Hyperperfect numbers* are another generalization of perfect
numbers, not to be confused with the better known multiply
perfect, multiperfect, or *k*-fold perfect numbers.

**Definition.**
An integer *n* > 1 is *k*-*hyperperfect* if it is
1 more than *k* times the sum of its proper divisors,
for some positive integer *k* called the
*index of perfection*.
(See Guy, section B2; Roberts, page 177;
Weisstein;
Sloane, sequences A007592,
A034897,
A007593,
A007594, etc.;
Sloane and Plouffe, sequences M4150, M5113, M5121.)

This is equivalent to

n=k(sigma(n)-n-1)+1(1)

where *sigma* is the usual sum of divisors function.

**Notation.**
Unless otherwise noted, *n* denotes a hyperperfect number,
*k* the index of perfection,
*p*, *q* and *r* are odd primes
with *p*<*q*<*r*, and *i* and *k* are
positive integers.

All hyperperfect numbers less than 10^{11} have been tabulated in
this study. There are 2190 hyperperfect numbers in this range, for 1932
different values of *k*. Only 85 of the hyperperfect numbers
have odd index *k*,
and 80 distinct odd values of *k* are represented. A
total of 2105 of the hyperperfect numbers have even index *k*, and
1852 distinct even values of *k* are represented. All of these
hyperperfect numbers are odd except for the 1-hyperperfect numbers
(the familiar perfect numbers).
Some individual larger hyperperfect numbers are given later.

Table 1 is a list of the hyperperfect numbers less than 1,000,000 and their
index of perfection *k*. Sequence A034897
is the left column and A034898
is the right column. (Omitting the entries with *k*=1 gives
A007592.)

n |
k |
---|---|

6 | 1 |

21 | 2 |

28 | 1 |

301 | 6 |

325 | 3 |

496 | 1 |

697 | 12 |

1333 | 18 |

1909 | 18 |

2041 | 12 |

2133 | 2 |

3901 | 30 |

8128 | 1 |

10693 | 11 |

16513 | 6 |

19521 | 2 |

24601 | 60 |

26977 | 48 |

51301 | 19 |

96361 | 132 |

130153 | 132 |

159841 | 10 |

163201 | 192 |

176661 | 2 |

214273 | 31 |

250321 | 168 |

275833 | 108 |

296341 | 66 |

306181 | 35 |

389593 | 252 |

486877 | 78 |

495529 | 132 |

542413 | 342 |

808861 | 366 |

Table 2 is a list of the known hyperperfect numbers with *k* <=
100. The smallest known hyperperfect number for each value of *k* yields
sequence A007594.
Hyperperfect numbers less than 10^{11} are listed. Where there is no
hyperperfect number less than 10^{11}, and larger hyperperfect numbers
for this value of *k* are known, see Table 7.

k |
k-hyperperfect numbers |
---|---|

1 | 6, 28, 496, 8128, et al - the perfect numbers (A000396) |

2 | 21, 2133, 19521, 176661, 129127041 (A007593) |

3 | 325 |

4 | 1950625, 1220640625 |

6 | 301, 16513, 60110701, 1977225901 (A028499) |

10 | 159841 |

11 | 10693 |

12 | 697, 2041, 1570153, 62722153, 10604156641, 13544168521 (A028500) |

16 | see Table 7 |

18 | 1333, 1909, 2469601, 893748277 (A028501) |

19 | 51301 |

22 | see Table 7 |

28 | see Table 7 |

30 | 3901, 28600321 |

31 | 214273 |

35 | 306181 |

36 | see Table 7 |

40 | 115788961 |

42 | see Table 7 |

46 | see Table 7 |

48 | 26977, 9560844577 |

52 | see Table 7 |

58 | see Table 7 |

59 | 1433701 |

60 | 24601 |

66 | 296341 |

72 | see Table 7 |

75 | 2924101 |

78 | 486877 |

88 | see Table 7 |

91 | 5199013 |

96 | see Table 7 |

100 | 10509080401 |

We consider the cases of even *k* and odd *k* separately.

**Case 1: odd values of **** k.**
When

**Theorem 1.** If *k*>1 is an odd integer,
*p*=(3*k*+1)/2 is prime, and
*q*=3*k*+4=2*p*+3 is prime then
*p*^{2}*q* is *k*-hyperperfect.

**Proof.**
The proofs of Theorems 1, 2 and 3 are straightforward verifications and will be omitted.

An equivalent formulation is *p*=6*i*-1,
*q*=12*i*+1, and *k*=4*i*-1, for some
*i*>0. The proof does not hold if *p* is not of this form.

Theorem 1 also holds for *k*=1, giving the perfect number 28. Of
course, the other 1-hyperperfect numbers are not of that form.

For odd *k*>1, there are 79 *k*-hyperperfect numbers less
than 10^{11}. The smallest is 325 = 5^{2}*13, which is
3-hyperperfect. The largest of these is 98015605201 = 3659^{2}*7321,
which is 2439-hyperperfect.

Sequences A034934,
A034936,
A034937,
A034938,
A002476
and A045309
give primes related to Theorem 1. Table 3 lists odd values of *k*>1
for which there are *k*-hyperperfect numbers. All
(in fact all known *k*-hyperperfect numbers for odd *k*>1) are of the form of
Theorem 1 (sequence A038536):

Odd values of k having k-hyperperfect numbers |
---|

3, 11, 19, 31, 35, 59, 75, 91, 111, 115, 131, 151, 179, 235, 255, 311, 335, 339, 371, 375, 399, 411, 431, 439, 495, 515, 531, 539, 551, 591, 619, 675, 739, 791, 795, 811, 839, 851, 871, 915, 951, 999, 1015, 1035, 1039, 1055, 1071, 1075, 1155, 1231, 1351, 1375, 1391, 1399, 1419, 1515, 1531, 1539, 1595, 1599, 1651, 1699, 1851, 1859, 1879, 1895, 1939, 1951, 1959, 2091, 2111, 2139, 2219, 2259, 2275, 2351, 2355, 2411, 2439 |

**Conjecture 1 (Converse of Theorem 1).** All
*k*-hyperperfect numbers for odd *k*>1 are of the form given in
Theorem 1.

If *n* is a *k*-hyperperfect number for even *k*>1
then clearly *n* is odd. All known *k*-hyperperfect numbers for
odd *k*>1 are odd. If Conjecture 1 holds, then all
*k*-hyperperfect numbers for *k*>1 are odd.

Herman te Riele [1981] noted that the six hyperperfect numbers for odd
*k* known at that time [Minoli, 1980] were all of a form equivalent to
that in Theorem 1.

**Case 2: even values of ***k*>1

**Theorem 2.** If *p* and *q* are distinct odd
primes such that *k*(*p*+*q*)=*pq*-1 for some
integer *k*, then *n*=*pq* is *k*-hyperperfect.
Equivalently, *q=(kp+1)/(p-k).*

Again we omit the proof.

There are some limitations on the values of *k*, *p*, and
*q* that satisfy Theorem 2: (a)
*k*<*p*<2*k*<*q*; and (b) except for
*k*=2 (where *p*=3, *q*=7), *p* and *q* are
congruent modulo 12, and *k* is a multiple of 6.

Table 4 gives some values of *p*, *q*, and *k* that
satisfy Theorem 2. More values of *p* are given in sequence A034913,
and values of *p* and *q* combined, in order, are contained in
sequence A034914.

p |
q |
k |
---|---|---|

3 | 7 | 2 |

7 | 43 | 6 |

13 | 157 | 12 |

17 | 41 | 12 |

23 | 83 | 18 |

31 | 43 | 18 |

47 | 83 | 30 |

53 | 509 | 48 |

67 | 4423 | 66 |

73 | 337 | 60 |

79 | 6163 | 78 |

113 | 2441 | 108 |

137 | 3617 | 132 |

139 | 19183 | 138 |

151 | 22651 | 150 |

157 | 829 | 132 |

163 | 26407 | 162 |

173 | 557 | 132 |

173 | 5813 | 168 |

193 | 1297 | 168 |

193 | 37057 | 192 |

**Theorem 3.** Suppose *k*>0 and *p*=*k*+1 is prime.
If
*q*=*p ^{i}* -

Note that when *k*=1 and *p*=2 the theorem gives the familiar
perfect numbers. Table 5 lists some examples of this theorem. Sequence A034915
gives the values of *q* in order.

p |
q |
i |
---|---|---|

2 | 3 | 2 |

2 | 7 | 3 |

2 | 31 | 5 |

2 | 127 | 7 |

2 | 8191 | 13 |

2 | 131071 | 17 |

2 | 524287 | 19 |

3 | 7 | 2 |

3 | 79 | 4 |

3 | 241 | 5 |

3 | 727 | 6 |

3 | 19681 | 9 |

5 | 3121 | 5 |

5 | 78121 | 7 |

7 | 43 | 2 |

7 | 337 | 3 |

7 | 117643 | 6 |

7 | 40353601 | 9 |

11 | 1321 | 3 |

13 | 157 | 2 |

13 | 28549 | 4 |

13 | 371281 | 5 |

13 | 4826797 | 6 |

19 | 6841 | 3 |

19 | 130303 | 4 |

19 | 2476081 | 5 |

31 | 29761 | 3 |

31 | 28629121 | 5 |

41 | 68881 | 3 |

41 | 115856161 | 5 |

43 | 3418759 | 4 |

47 | 229344961 | 5 |

61 | 844596241 | 5 |

67 | 4423 | 2 |

79 | 6163 | 2 |

79 | 38950003 | 4 |

For convenience, we will say hyperperfect numbers produced by
Theorems 1, 2 and 3 are of *forms* 1, 2 and 3, respectively.
Minoli
[1980] gave a different (broader) sufficient condition for a number to be
hyperperfect, which is also necessary for hyperperfect numbers of the form
*p ^{i}q* and does not depend on the parity of

For even *k*>1, there are 2105 *k*- hyperperfect numbers
less than 10^{11}. The smallest of these is 21, which is 2-hyperperfect.
The largest is 99671702281=107693*925517, which is 6468-hyperperfect. The
largest even value of *k* represented is 156102, where 97885007917 =
293147*333911 is 156102-hyperperfect. Of these 2105 hyperperfect numbers, 2001
are of form 2 only, 17 are of form 3 only, 68 are of both forms, and 19 are of
neither form. The known hyperperfect numbers that don't fit these forms all
have three distinct prime factors. Thus all known hyperperfect numbers of
the form *p ^{i}q* are of forms 1, 2 or 3.
The largest hyperperfect number less than 10

Table 6 gives the hyperperfect numbers less than 10^{11} that are of
form 3 but not of form 2:

n |
k |
factorization of n |
form of q |
---|---|---|---|

2133 | 2 | 3^{3} 79 |
3^{4}-3+1 |

16513 | 6 | 7^{2} 337 |
7^{3}-7+1 |

19521 | 2 | 3^{4} 241 |
3^{5}-3+1 |

159841 | 10 | 11^{2} 1321 |
11^{3}-11+1 |

176661 | 2 | 3^{5} 727 |
3^{6}-3+1 |

1950625 | 4 | 5^{4} 3121 |
5^{5}-5+1 |

2469601 | 18 | 19^{2} 6841 |
19^{3}-19+1 |

28600321 | 30 | 31^{2} 29761 |
31^{3}-31+1 |

62722153 | 12 | 13^{3} 28549 |
13^{4}-13+1 |

115788961 | 40 | 41^{2} 68881 |
41^{3}-41+1 |

129127041 | 2 | 3^{8} 19681 |
3^{9}-3+1 |

893748277 | 18 | 19^{3} 130303 |
19^{4}-19+1 |

1220640625 | 4 | 5^{6} 78121 |
5^{7}-5+1 |

1977225901 | 6 | 7^{5} 117643 |
7^{6}-7+1 |

10509080401 | 100 | 101^{2} 1030201 |
101^{3}-101+1 |

10604156641 | 12 | 13^{4} 371281 |
13^{5}-13+1 |

51886178401 | 138 | 139^{2} 2685481 |
139^{3}-139+1 |

For even values of *k* for which *k*-hyperperfect numbers exist, it is
more common for there to be *k*-hyperperfect numbers when *k* is a
multiple of 6 (form 2). For the 1852 even values of *k* having a
*k*-hyperperfect number less than 10^{11}, all are multiples of 6
except for *k* = 2, 4, 10, 40, 100, 140, and 190. The first five of these
cases have *k*+1 prime, and thus are hyperperfect numbers of form 3. For
the other two cases, 157*2131*3343 is 140-hyperperfect and 229*1999*2551 is
190-hyperperfect.

We can apply Theorem 3 to find some large *k*-hyperperfect numbers
when *k*+1=*p* is prime. For instance, referring to Table 2; there
are no small (i.e. < 10^{11} ) *k*-hyperperfect numbers for
*k*=16, 22, 28, 36, 42, 46, etc - cases in which *k*+1 is prime.
(There are other small values such as *k*=8, in which no 8-hyperperfect
numbers are known.) We only have to check to see if *q=p ^{i}-p*
+1 is prime for some

k |
p |
values of i resulting in primes |
---|---|---|

16 | 17 | 11, 21, 127, 149, 469 (A034922) |

22 | 23 | 17, 61, 445 |

28 | 29 | 33, 89, 101 |

36 | 37 | 67, 95, 341 |

42 | 43 | 4, 6, 42, 64, 65 (A034923) |

46 | 47 | 5, 11, 13, 53, 115 (A034924) |

52 | 53 | 21, 173 |

58 | 59 | 11, 117 |

70 | 71 | none |

72 | 73 | 21, 49 |

82 | 83 | none |

88 | 89 | 9, 41, 51, 109, 483 (A034925) |

96 | 97 | 6, 11, 34 |

100 | 101 | 3, 7, 9, 19, 29, 99, 145 (A034926) |

Table 7 fills in some of the values for *k*<=100 in Table 2 for
which there are no hyperperfect numbers < 10^{11}. A method was given
by te Riele [1981] for generating hyperperfect numbers with three or more factors.
He also gave hyperperfect numbers for *k* = 42, 72, and 96. A computation
using this method (except not requiring *p*=*k*+1) for
*p*<2^{16}, *q*<*r*< 2^{31} did
not reveal any additional hyperperfect numbers for *k*<=100.

A corollary of the prime number theorem is that the probability that a given
integer *x* is prime is approximately 1/*ln*(*x*).
Considering numbers of form 3, the probability that *q* is prime is
approximately 1/*ln*(*p ^{i}*). Since the sum of this
quantity for

Nineteen of the hyperperfect numbers less than 10^{11} have three
distinct prime factors (the first prime factor may be to a power greater than
one) and none of them have more than three distinct factors. For even values of
*k*, seventeen examples are of the form *p ^{i}q* , for

n |
k |
factorization of n |
source |
---|---|---|---|

1570153 | 12 | 13 269 449 | te Riele |

60110701 | 6 | 7^{2} 383 3203 |
te Riele |

391854937 | 228 | 547 569 1259 | |

1118457481 | 140 | 157 2131 3343 | |

1167773821 | 190 | 229 1999 2551 | |

1218260233 | 252 | 349 1481 2357 | |

1564317613 | 198 | 373 443 9467 | |

2469439417 | 372 | 677 1103 3307 | |

6287557453 | 438 | 733 1307 6563 | |

8942902453 | 402 | 547 1831 8929 | |

9560844577 | 48 | 61 229 684433 | |

12161963773 | 126 | 191 373 170711 | |

13544168521 | 12 | 13^{2} 2347 34147 |
te Riele |

23911458481 | 360 | 659 809 44851 | |

26199602893 | 342 | 661 719 55127 | |

31571188513 | 816 | 1493 2221 9521 | |

46727970517 | 138 | 229 349 584677 | |

64169172901 | 1050 | 1831 3169 11059 | |

80293806421 | 1410 | 3491 4073 5647 |

A search was made for hyperperfect numbers of the form *pqr* using the
method of te Riele [1981], except not requiring that *p*=*k*+1 (as
he did for practical reasons). This search was restricted to *k* <=
10,000 and *p*-*k* <= 1000. An additional 346 hyperperfect
numbers of the form *n*=*pqr*, *n*>10^{11} were
found. The largest value of *k* was 9930, for which
10009*1258219*125066187236071 is 9330-hyperperfect. Table 9 lists the ones found
for *k* <= 1000.

k |
p |
q |
r |
---|---|---|---|

12 | 13 | 269 | 449 |

48 | 61 | 229 | 684433 |

126 | 191 | 373 | 170711 |

136 | 193 | 463 | 1748863 |

138 | 229 | 349 | 584677 |

140 | 157 | 2131 | 3343 |

174 | 211 | 997 | 36814051 |

180 | 211 | 1231 | 47012941 |

190 | 229 | 1999 | 2551 |

192 | 197 | 8369 | 83101 |

198 | 373 | 443 | 9467 |

206 | 211 | 8737 | 29354287 |

206 | 211 | 8971 | 331213 |

222 | 223 | 49807 | 31352557 |

228 | 229 | 67187 | 238919 |

228 | 263 | 1733 | 225427 |

228 | 547 | 569 | 1259 |

252 | 349 | 1481 | 2357 |

276 | 277 | 78541 | 3323977 |

282 | 283 | 112087 | 280537 |

296 | 463 | 823 | 1166713 |

342 | 661 | 719 | 55127 |

348 | 349 | 133183 | 1425091 |

350 | 541 | 997 | 260413 |

360 | 659 | 809 | 44851 |

372 | 677 | 1103 | 3307 |

396 | 601 | 1163 | 12064691 |

402 | 421 | 8929 | 216417217 |

402 | 547 | 1831 | 8929 |

408 | 419 | 17123 | 172681 |

414 | 641 | 1171 | 10741487 |

430 | 433 | 63067 | 4560151 |

438 | 733 | 1307 | 6563 |

480 | 613 | 2221 | 973057 |

522 | 523 | 273629 | 741044219 |

522 | 823 | 1429 | 615082519 |

546 | 547 | 471677 | 818291 |

570 | 571 | 329519 | 30881489 |

570 | 937 | 1459 | 984367 |

660 | 911 | 2399 | 6308329 |

672 | 673 | 453367 | 467751847 |

684 | 757 | 12791 | 15971 |

774 | 821 | 13537 | 783023081 |

810 | 887 | 9473 | 671971 |

816 | 1493 | 2221 | 9521 |

820 | 823 | 234319 | 5804353 |

968 | 1123 | 7027 | 6631993 |

972 | 977 | 221707 | 1334603 |

978 | 1031 | 19163 | 3049369 |

Herman te Riele constructed eleven hyperperfect numbers with three distinct
prime factors and one with four distinct prime factors. In his examples with
three prime factors, he set *p*=*k*+1 for practical reasons; but
that restriction is not necessary. This survey found sixteen additional
hyperperfect numbers less than 10^{11} with three prime factors. The
numbers that te Riele constructed that are less than 10^{11} are noted
above. Table 10 lists hyperperfect numbers (for even k) with a prime factor to
higher than first power:

n |
k |
factorization of n |
---|---|---|

2133 | 2 | 3^{3} 79 |

16513 | 6 | 7^{2} 337 |

19521 | 2 | 3^{4} 241 |

159841 | 10 | 11^{2} 1321 |

176661 | 2 | 3^{5} 727 |

1950625 | 4 | 5^{4} 3121 |

2469601 | 18 | 19^{2} 6841 |

28600321 | 30 | 31^{2} 29761 |

60110701 | 6 | 7^{2} 383 3203 |

62722153 | 12 | 13^{3} 28549 |

115788961 | 40 | 41^{2} 68881 |

129127041 | 2 | 3^{8} 19681 |

893748277 | 18 | 19^{3} 130303 |

1220640625 | 4 | 5^{6} 78121 |

1977225901 | 6 | 7^{5} 117643 |

10509080401 | 100 | 101^{2} 1030201 |

10604156641 | 12 | 13^{4} 371281 |

13544168521 | 12 | 13^{2} 2347 34147 |

51886178401 | 138 | 139^{2} 2685481 |

The method of te Riele can not yield *k*-hyperperfect numbers of the
form *pqr* for odd *k*. In that construction,
*n*/*p* is even except when *k*=1 and *p*=2, so
*n*/*p* cannot be factored into odd primes *q* and
*r*.

Let us examine some small values of *k*. For *k*=2 all five
examples are of form 3, as are both examples for *k*=4 and three of the
four examples for *k*=6, the example for *k*=10, and others. The
examples that are not of form 2 or form 3 can be constructed by the method of te
Riele.
Table 11 gives some examples with small *k*, which tend to be of form 3:

n |
k |
factorization of n |
form |
---|---|---|---|

21 | 2 | 3 7 | form 2 and form 3 |

2133 | 2 | 3^{3} 79 |
form 3 |

19521 | 2 | 3^{4} 241 |
form 3 |

176661 | 2 | 3^{5} 727 |
form 3 |

129127041 | 2 | 3^{8} 19681 |
form 3 |

1950625 | 4 | 5^{4} 3121 |
form 3 |

1220640625 | 4 | 5^{6} 78121 |
form 3 |

301 | 6 | 7 43 | form 2 and form 3 |

16513 | 6 | 7^{2} 337 |
form 3 |

60110701 | 6 | 7^{2} 383 3203 |
te Riele construction |

1977225901 | 6 | 7^{5} 117643 |
form 3 |

159841 | 10 | 11^{2} 1321 |
form 3 |

697 | 12 | 17 41 | form 2 |

2041 | 12 | 13 157 | form 2 and form 3 |

1570153 | 12 | 13 269 449 | te Riele construction |

62722153 | 12 | 13^{3} 28549 |
form 3 |

10604156641 | 12 | 13^{4} 371281 |
form 3 |

13544168521 | 12 | 13^{2} 2347 34147 |
te Riele construction |

(26-digit #) | 16 | 17^{10} ( 17^{11}-17+1 ) |
form 3 |

1333 | 18 | 31 43 | form 2 |

1909 | 18 | 23 83 | form 2 |

2469601 | 18 | 19^{2} 6841 |
form 3 |

893748277 | 18 | 19^{3} 130303 |
form 3 |

Several values of *k* in table 11 have multiple
*k*-hyperperfect numbers. Table 12 lists some examples with large *k
*that are represented by several hyperperfect numbers, all of which are of
form 2.

n |
k |
factorization of n |
---|---|---|

4660241041 | 31752 | 46457 100313 |

7220722321 | 31752 | 38153 189257 |

12994506001 | 31752 | 34693 374557 |

52929885457 | 31752 | 32381 1634597 |

60771359377 | 31752 | 32297 1881641 |

15166641361 | 55848 | 78593 192977 |

44783952721 | 55848 | 60397 741493 |

67623550801 | 55848 | 58693 1152157 |

18407557741 | 67782 | 130307 141263 |

18444431149 | 67782 | 127867 144247 |

34939858669 | 67782 | 80287 435187 |

50611924273 | 92568 | 118061 428693 |

64781493169 | 92568 | 109793 590033 |

84213367729 | 92568 | 104593 805153 |

50969246953 | 100932 | 139429 365557 |

53192980777 | 100932 | 136057 390961 |

82145123113 | 100932 | 118057 695809 |

For 204 values of *k*, there are two or more *k*-hyperperfect numbers
less than 10^{11}. Values of *k* with more than three examples
are shown in table 13:

k |
# | terms (sequence) |
---|---|---|

1 | 6 | 6, 28, 496, 8128, 33550336, 8589869056 (A000396) |

2 | 5 | 21, 2133, 19521, 176661, 129127041 (A007593 ) |

6 | 4 | 301, 16513, 60110701, 1977225901 (A028499) |

12 | 6 | 697, 2041, 1570153, 62722153, 10604156641, 13544168521 (A028500) |

18 | 4 | 1333, 1909, 2469601, 893748277 (A028501) |

2772 | 4 | 95295817, 124035913, 749931337, 4275383113 (A028502) |

3918 | 4 | 61442077, 217033693, 12059549149, 60174845917 |

9222 | 4 | 404458477, 3426618541, 8983131757, 13027827181 |

9828 | 4 | 432373033, 2797540201, 3777981481, 13197765673 |

14280 | 4 | 848374801, 2324355601, 4390957201, 16498569361 |

23730 | 4 | 2288948341, 3102982261, 6861054901, 30897836341 |

31752 | 5 | 4660241041, 7220722321, 12994506001, 52929885457, 60771359377 (A034916) |

In view of Theorem 3, there should be *k*-hyperperfect numbers
whenever *k*+1 is prime. When *k* is even and *k*+1 is
composite the situation is less clear. For a value of *k* that is a
multiple of 6, Theorem 2 provides only a finite number of possible
*k*-hyperperfect numbers. The search up to 10^{11} revealed
hyperperfect numbers for some of these values of *k*, but Theorem 2 fails
to provide any more examples. Therefore there are even values of *k* for
which (a) there are no *k*-hyperperfect numbers less than
10^{11}, (b) Theorem 2 fails to provide any examples, and (c) Theorem 3
does not apply. However, there could be hyperperfect numbers larger than
10^{11} of different forms for these even values of *k*. For
example, 157*2131*3343 is 140-hyperperfect and 229*1999*2551 is
190-hyperperfect.

Daniel Minoli and Robert Bear [Guy, section B2] conjectured that there
are *k*-hyperperfect numbers for every *k*. The data presented
here can be taken as evidence that this conjecture is false. The most compelling
reason is that the data suggests that the converse of Theorem 1 (Conjecture 1)
is true, which would mean that there are odd values of *k* for which
there are no *k*-hyperperfect numbers. Furthermore, as noted before, te
Riele's construction (with three or more prime factors) is inapplicable for odd
*k*.

For even values of *k* the situation is less clear. There are even
values of *k* for which no *k*-hyperperfect number is known. If
*k*+1 is prime then Theorem 3 should eventually produce a
*k*-hyperperfect number. If *k* is a multiple of 6 then theorem 2
provides only a finite number of possibilities. Otherwise there is a chance that
the method of te Riele will generate an example. However this chance seems
small, and hyperperfect numbers constructed this way are rare. Considering the
foregoing, the following conjecture is offered:

**Conjecture 2.** There are even values of *k* for which
there are no *k*-hyperperfect numbers.

For odd values of *k*>1 we have given a construction which produces
*k*-hyperperfect number, and we conjecture that all such hyperperfect
numbers are of this form (for odd *k*>1).

For even values of *k*, we have exhibited two sufficient conditions that
result in *k*-hyperperfect numbers.
All known
hyperperfect numbers with exactly two distinct prime factors are
one of these two forms, but hyperperfect numbers with more than two distinct prime
factors exist which are not of these forms.
Some of these numbers were also constructed by te
Riele.

We have given some evidence arguing against the conjecture published by Minoli
and Bear that *k*-hyperperfect numbers exist for all *k*>0.

A final note: Minoli [1980] gave a list of the hyperperfect numbers
less than 1,500,000 and stated that the computation took over ten hours of time
on a PDP 11/70. This author's program searched the same range in under six
seconds on a 300 MHz Pentium-II general-purpose electronic computer.
Searching up to 10^{11} required several overnight runs, however.

A034897 - Hyperperfect numbers (including 1-hyperperfect)

A034898 - Index of perfection of the terms of A034897

A007594/M4150 - Smallest

A038536 - Odd values of

Primes related to hyperperfect numbers of certain forms:

A034934,
A034936,
A034937,
A034938,
A002476,
A045309
- form 1

A034913,
A034914
- form 2, Table 4

A034915
- form 3, Table 5

A034922,
A034923,
A034924,
A034925,
A034926
- form 3, Table 7

Some values of *k* with at least four known *k*-hyperperfect
numbers:

A000396/M4186
- Perfect numbers, 1-hyperperfect numbers

A007593/M5121
- 2-hyperperfect numbers (5 known)

A028499
- 6-hyperperfect numbers (4 known)

A028500
- 12-hyperperfect numbers (6 known)

A028501
- 18-hyperperfect numbers (4 known)

A028502
- 2772-hyperperfect numbers (4 known)

A034916
- 31752-hyperperfect numbers (5 known)

Richard K. Guy, *Unsolved Problems in Number Theory*, second edition,
Springer-Verlag, New York, 1994.

Daniel Minoli, Issues in nonlinear hyperperfect numbers, *Mathematics of
Computation*, vol. 34, 639-645, 1980.

Joe Roberts, *Lure of the Integers*, Mathematical Association of
America, 1992. (Note: the definition of hyperperfect on page 177 contains a
misprint: "*sigma*(*n*)" should be "*sigma*(*m*)".)

Herman J. J. te Riele, Hyperperfect numbers with three different prime
factors, *Mathematics of Computation*, vol. 36, 297-298, 1981.

N. J. A. Sloane, *The On-Line Encyclopedia of Integer Sequences*,
published electronically at
www.research.att.com/~njas/sequences/

N. J. A. Sloane and Simon Plouffe, *The Encyclopedia of Integer
Sequences*, Academic Press, San Diego, 1995.

Eric W. Weisstein, *The CRC Concise Encyclopedia of Mathematics, *CRC
Press, Cleveland, 1998. Online version:
mathworld.wolfram.com/

(Concerned with sequences A007592, A007593, A007594, A028499, A028500, A028501, A028502, A038536, A034897, A034898, A034913, A034914, A034915, A034916, A034922, A034923, A034924, A034925, A034926, A034934, A034936, A034937, A034938. )

Received August 4, 1998; revised version received October 22, 1999. Published in Journal of Integer Sequences Jan. 21, 2000.

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