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

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 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.

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):

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<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.

Theorem 3. Suppose k>0 and p=k+1 is prime. If q=pⁱ -p+1 is prime for some i>1 then n=p^{i -1}q 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.

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 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:

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 p^{i -1}q 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 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 [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<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:

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>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.

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:

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:

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.

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:

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 [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 10¹¹ required several overnight runs, however.

7. Relevant sequences

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.