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

## A Study of Hyperperfect Numbers

### Judson S. McCranie 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 1011 have been computed. Evidence is presented suggesting that a published conjecture is false.

## 1. Introduction

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 1011 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.

## 2. The main tables

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.)

Table 1.
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 1011 are listed. Where there is no hyperperfect number less than 1011, and larger hyperperfect numbers for this value of k are known, see Table 7.

Table 2.
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

## 3. Constructions

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

Case 1: odd values of k. When k=1 these are the perfect numbers, and we will say no more about them. For the remainder of this section, we consider odd k>1, unless noted otherwise.

Theorem 1. If k>1 is an odd integer, p=(3k+1)/2 is prime, and q=3k+4=2p+3 is prime then p2q is k-hyperperfect.

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

An equivalent formulation is p=6i-1, q=12i+1, and k=4i-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 1011. The smallest is 325 = 52*13, which is 3-hyperperfect. The largest of these is 98015605201 = 36592*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):

Table 3.
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  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<2k<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.

Table 4.
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=pi -p+1 is prime for some i>1 then n=pi -1q is k-hyperperfect.

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.

Table 5.
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  gave a different (broader) sufficient condition for a number to be hyperperfect, which is also necessary for hyperperfect numbers of the form piq and does not depend on the parity of k.

For even k>1, there are 2105 k- hyperperfect numbers less than 1011. 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 piq are of forms 1, 2 or 3. The largest hyperperfect number less than 1011 of form 3 is also of form 2: 94860412321 = 4561*20798161 = pq; k=4560.

Table 6 gives the hyperperfect numbers less than 1011 that are of form 3 but not of form 2:

Table 6.
n k factorization of n form of q
2133 2 33 79 34-3+1
16513 6 72 337 73-7+1
19521 2 34 241 35-3+1
159841 10 112 1321 113-11+1
176661 2 35 727 36-3+1
1950625 4 54 3121 55-5+1
2469601 18 192 6841 193-19+1
28600321 30 312 29761 313-31+1
62722153 12 133 28549 134-13+1
115788961 40 412 68881 413-41+1
129127041 2 38 19681 39-3+1
893748277 18 193 130303 194-19+1
1220640625 4 56 78121 57-5+1
1977225901 6 75 117643 76-7+1
10509080401 100 1012 1030201 1013-101+1
10604156641 12 134 371281 135-13+1
51886178401 138 1392 2685481 1393-139+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 1011, 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. < 1011 ) 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=pi-p +1 is prime for some i>1 - if so then pi -1q is hyperperfect by Theorem 3. Table 7 shows the large hyperperfect numbers were found for k<=100, k+1=p prime, and i<=500:

Table 7.
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 < 1011. A method was given by te Riele  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<216, q<r< 231 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(pi). Since the sum of this quantity for i from 2 to infinity diverges, we expect an infinite number of k-hyperperfect numbers when k+1 is prime.

## 4. More than two primes

Nineteen of the hyperperfect numbers less than 1011 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 piq , for i>1, p<q, whereas 2069 of the examples are of the form pq, and two are of the form piqr . Table 8 gives hyperperfect numbers less than 1011 with more than two distinct prime factors:

Table 8.
n k factorization of n source
1570153 12 13 269 449 te Riele
60110701 6 72 383 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 12 132 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 , 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>1011 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.

Table 9.
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 1011 with three prime factors. The numbers that te Riele constructed that are less than 1011 are noted above. Table 10 lists hyperperfect numbers (for even k) with a prime factor to higher than first power:

Table 10.
n k factorization of n
2133 2 33 79
16513 6 72 337
19521 2 34 241
159841 10 112 1321
176661 2 35 727
1950625 4 54 3121
2469601 18 192 6841
28600321 30 312 29761
60110701 6 72 383 3203
62722153 12 133 28549
115788961 40 412 68881
129127041 2 38 19681
893748277 18 193 130303
1220640625 4 56 78121
1977225901 6 75 117643
10509080401 100 1012 1030201
10604156641 12 134 371281
13544168521 12 132 2347 34147
51886178401 138 1392 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:

Table 11.
n k factorization of n form
21 2 3 7 form 2 and form 3
2133 2 33 79 form 3
19521 2 34 241 form 3
176661 2 35 727 form 3
129127041 2 38 19681 form 3
1950625 4 54 3121 form 3
1220640625 4 56 78121 form 3
301 6 7 43 form 2 and form 3
16513 6 72 337 form 3
60110701 6 72 383 3203 te Riele construction
1977225901 6 75 117643 form 3
159841 10 112 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 12 133 28549 form 3
10604156641 12 134 371281 form 3
13544168521 12 132 2347 34147 te Riele construction
(26-digit #) 16 1710 ( 1711-17+1 ) form 3
1333 18 31 43 form 2
1909 18 23 83 form 2
2469601 18 192 6841 form 3
893748277 18 193 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.

Table 12.
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

## 5. Remarks about general values of k

For 204 values of k, there are two or more k-hyperperfect numbers less than 1011. Values of k with more than three examples are shown in table 13:

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 1011 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 1011, (b) Theorem 2 fails to provide any examples, and (c) Theorem 3 does not apply. However, there could be hyperperfect numbers larger than 1011 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.

## 6. Conclusions

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  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 1011 required several overnight runs, however.

## 7. Relevant sequences

A007592/M5113 - Hyperperfect numbers (35 numbers up to 1232053, omitting perfect numbers)
A034897 - Hyperperfect numbers (including 1-hyperperfect)
A034898 - Index of perfection of the terms of A034897
A007594/M4150 - Smallest k-hyperperfect number (some terms believed not to exist)
A038536 - Odd values of k with hyperperfect numbers

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)

## References

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.