Primitive weird numbers having more than three distinct prime factors
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton
aa r X i v : . [ m a t h . N T ] M a r PRIMITIVE WEIRD NUMBERS HAVING MORE THAN THREE DISTINCT PRIMEFACTORS
GIANLUCA AMATO, MAXIMILIAN F. HASLER, GIUSEPPE MELFI, AND MAURIZIO PARTON
Abstract.
In this paper we study some structure properties of primitive weird numbers in terms of theirfactorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms forgenerating weird numbers having a given number of distinct prime factors are presented. These algorithmsyield primitive weird numbers of the form mp . . . p k for a suitable deficient positive integer m and primes p , . . . , p k and generalize a recent technique developed for generating primitive weird numbers of the form2 n p p . The same techniques can be used to search for odd weird numbers, whose existence is still an openquestion. Introduction
Let n ∈ N be a natural number, and let σ ( n ) = P d | n d be the sum of its divisors. If σ ( n ) > n , then n is called abundant , whereas if σ ( n ) < n , then n is called deficient . Perfect numbers are those for which σ ( n ) = 2 n .According to [5], we will refer to ∆( n ) = σ ( n ) − n as the abundance of n , and to d ( n ) = 2 n − σ ( n ) = − ∆( n ) asthe deficience of n . If n can be expressed as a sum of distinct proper divisors, then n is called semiperfect , orsometimes also pseudoperfect . Slightly abundant numbers with ∆( n ) = 1 are called quasi-perfect , and slightlydeficient numbers with d ( n ) = 1 are called almost perfect .A weird number is a number which is abundant but not semiperfect. In other words, n ∈ N is weird if it isabundant and it cannot be written as the sum of some of its proper divisors.Weird numbers have been defined in 1972 by Benkoski [1], and appear to be rare: for instance, up to 10 wehave only 7 of them [10]. Despite this apparent rarity, which is the reason of the name, weird numbers areeasily proven to be infinite: if n is weird and p is a prime larger than σ ( n ), then np is weird (see for example[4, page 332]). But a much stronger property is true: Benkoski and Erd˝os, in their joint 1974 paper [2], provedthat the set of weird numbers has positive asymptotic density.Several questions on weird numbers have not been settled yet. For instance, if we look for primitive weirdnumbers , that is, that are not multiple of other weird numbers, we don’t know whether they are infinite or not: Conjecture 1.1. [2, end of page 621]
There exist infinitely many primitive weird numbers.
In this respect, the third author recently proved in [8] that the infiniteness of primitive weird numbers followsby assuming the classic Cram´er conjecture on gaps between consecutive primes [3].Another open question is the existence of odd weird numbers. Erd˝os offered $ 10 for an example of odd weirdnumber, and $ 25 for a proof that none can exist [1]. Recently Wenjie Fang [10, Sequence A006037] claimedthat there are no odd weird numbers up to 10 , and no odd weird numbers up to 10 with abundance notexceeding 10 .Moreover, very little is known about a pattern in the prime factorization of primitive weird numbers. As ofnow, most of the known primitive weird numbers are of the form 2 n pq with p and q primes, and all the paperson primitive weird numbers deal with numbers of this form [6, 7, 8, 9]. Relatively few examples of primitiveweird numbers with more than three distinct prime factors are known up to now: for instance among the 657primitive weird numbers not exceeding 1 . · there are 531 primitive weird numbers having three distinctprime factors; 69 having four distinct prime factors; 54 having five distinct prime factors and only 3 having sixdistinct prime factors. Mathematics Subject Classification.
Primary 11A25; Secondary 11B83.
Key words and phrases. abundant numbers, semiperfect numbers, almost perfect numbers, sum-of-divisor function, Erd˝osproblems, weird numbers, primitive weird numbers.Part of this research was done while the first author was visiting the Department of Mathematics and Computer Science atWesleyan University.
This paper considers primitive weird numbers that have several distinct prime factors. In particular, we givesufficient conditions in order to ensure that a positive integer of the form mp . . . p k is weird, where m is adeficient number and p , . . . , p k are primes (see Theorem 3.1).We then apply Theorem 3.1 to search for new primitive weird numbers, looking in particular at four or moreprime factors. We find hundreds of primitive weird numbers with four distinct prime factors of the form2 m p p p , 75 primitive weird numbers with five distinct prime factors of the form 2 m p p p p , and 9 primitiveweird numbers with six distinct prime factors (see Section 4).This paper generalizes to several factors a technique developed in [8]. This approach, as far as we know, is thefirst that can be used to generate primitive weird numbers with several distinct prime factors. Moreover, sincethere are many odd deficient numbers, Theorem 3.1 can be used to hunt for the first example of an odd weirdnumber (see Section 5). 2. Basic ideas
We recall a fundamental lemma that will be extensively used, and that corresponds to an equivalent definitionof weird number.
Lemma 2.1.
An abundant number w is weird if and only if ∆( w ) cannot be expressed as a sum of distinctdivisors of w .Proof. For a proof one can see [8, Lemma 2]. (cid:3)
We will need another technical lemma, which will be used in the proof of the main theorems.
Lemma 2.2. If w = mq is an abundant number, m is deficient, q is prime and q ≥ σ ( p α ) − for each p α || m ,then w is primitive abundant.Proof. Since a multiple of an abundant number is abundant, in order to prove that m is a primitive abundantnumber, (i.e., an abundant number whose proper divisors are all deficient), it suffices to prove that w/p isdeficient for each p | w . If p = q , then w/q = m , and we assumed that m is deficient. Otherwise, if p α || m , then σ ( w/p ) w/p = σ ( w )( p α − pw ( p α +1 −
1) = σ ( w ) w · (cid:18) − p − p α +1 − (cid:19) ≤ σ ( w ) w · (cid:18) − q + 1 (cid:19) = σ ( w ) w · qq + 1 = σ ( w/q ) w/q < . (cid:3) Main result
In this section we provide two ways for generating primitive weird numbers.
Theorem 3.1.
Let m > be a deficient number and k > . Let p , . . . , p k be primes with σ ( m ) + 1 < p < · · · < p k . Let h ∗ = (cid:20) p − σ ( m ) p k − p (cid:21) . Let ˜ w = mp . . . p k , and (1) U m,p ,p k := h ∗ [ j =0 { n ∈ N | jp k + σ ( m ) < n < ( j + 1) p } . If ˜ w is abundant and ∆( ˜ w ) ∈ U m,p ,p k , then w = ˜ w is a primitive weird number. If ˜ w is deficient, let p < wd ( ˜ w ) − a prime with p > p k . Then w = ˜ wp is abundant. Furthermore, if ∆( w ) ∈ U m,p ,p k , then p > wd ( ˜ w ) − − (1 + h ∗ ) p d ( ˜ w ) and moreover, if p > ∆( w ) then w is a primitive weird number. RIMITIVE WEIRD NUMBERS HAVING MORE THAN THREE DISTINCT PRIME FACTORS 3
Proof of 3.1.1 Since σ ( m ) + 2 ≤ p , the set union in the right side of (1) is not empty. The sets of consecutiveintegers involved in the union in the right side of (1) are pairwise disjoint. If h < p − σ ( m ) p k − p , then σ ( m ) + hp k < ( h + 1) p . Now, let j ≤ h ∗ and let n be an integer with jp k + σ ( m ) < n < ( j + 1) p . We will prove that n cannot beexpressible as a sum of distinct divisors of w .Note that n < p . This is because n < ( j + 1) p ≤ ( h ∗ + 1) p < ( p / p and p / < p . This meansthat if n is expressible as a sum of distinct divisors of w , these divisors must be of the form dp with d | m and p ∈ { p , . . . , p k } , or simply of the form d , with d | m . Let’s say n = d p + · · · + d N p N + d ′ + d ′ + · · · + d ′ M ,where d ′ , . . . , d ′ M are distinct divisors of m . Then, necessarily d + d + · · · + d N ≤ j , since n < ( j + 1) p . As aconsequence, we have: d ′ + · · · + d ′ M = n − ( d p + . . . d N p N ) ≥ n − jp k > σ ( m )and this is in contradiction with the assumption on d ′ , . . . , d ′ M . So n cannot be expressible as a sum of distinctdivisors of w .No elements in U m,p ,p k can be expressed as a sum of distinct divisors of w . Since ∆( w ) ∈ U m,p ,p k , byLemma 2.1 this implies that w is weird.In order to prove that w is a primitive weird number, by Lemma 2.2 it suffices to prove that w/p k is deficient.If k = 2, then ∆( w/p k ) = ∆( mp ) = ∆( m ) p + σ ( m ). Since ∆( m ) ≤ − p > σ ( m ), then ∆( w/p ) < k ≥ . Then we have:∆ (cid:18) wp k (cid:19) = σ (cid:18) wp k (cid:19) − wp k = σ ( w ) p k + 1 − wp k = p k ( σ ( w ) − w ) − wp k ( p k + 1) = ∆( w ) − wp k p k + 1Now, ∆( w ) ∈ U m,p ,p k . Since k ≥
3, between p and p k there is at least an odd integer that is not prime, andtherefore h ∗ < p / k . This means that∆( w ) < ( h ∗ + 1) p < (cid:16) p k (cid:17) p . On the other hand 2 w/p k ≥ mp , and since 2 mp > p + p > p / k , this means that ∆( w ) < w/p k andtherefore ∆( w/p k ) <
0. In particular, since w/p k is deficient, by Lemma 2.2, w is a primitive weird number. (cid:3) Proof of 3.1.2 Note that p is the largest prime that divides w , and that w/p = ˜ w is deficient. So by Lemma 2.2,in order to prove that w is a primitive weird number, it suffices to prove that w is indeed abundant and weird.Since ( ˜ w, p ) = 1 and 2 ˜ w − d ( ˜ w ) p − d ( ˜ w ) > w ) = σ ( w ) − w = σ ( ˜ w )( p + 1) − p ˜ w = ( σ ( ˜ w ) − w ) p + σ ( ˜ w )= − d ( ˜ w ) p + 2 ˜ w − d ( ˜ w ) > w is abundant.Now assume that ∆( w ) ∈ U m,p ,p k . Since max U m,p ,p k < (1 + h ∗ ) p , then ∆( w ) = − d ( ˜ w ) p + 2 ˜ w − d ( ˜ w ) < (1 + h ∗ ) p and therefore p > wd ( ˜ w ) − − (1 + h ∗ ) p d ( ˜ w )Let p > ∆( w ). In order to prove that w is weird, by Lemma 2.1, we have to prove that ∆( w ) is not a sum ofproper divisors of w .Since ∆( w ) < p , if ∆( w ) is a sum of proper divisors of w , then all divisors involved must be divisors of ˜ w . Onthe other hand ∆( w ) ∈ U m,p ,p k and as seen above, no element in U m,p ,p k can be expressed as a sum of distinctdivisors of ˜ w . This completes the proof. (cid:3) G. AMATO, M. F. HASLER, G. MELFI, AND M. PARTON
Remark . Very often, when conditions of Theorem 3.1.2 hold, it is (1 + h ∗ ) p < d ( ˜ w ) . This means that inthese cases, p = [2 ˜ w/d ( ˜ w ) − Remark . If m, p , . . . , p k is a sequence such that w = mp . . . p k is primitive weird according to Theo-rem 3.1.1, k > w ) < p k , then ˜ w = p . . . p k − and p = p k verify the conditions of Theorem 3.1.2Indeed, if w satisfies the condition of Theorem 3.1.1, then w is abundant and ˜ w = w/p k is deficient. Thisimplies that p k < w/d ( ˜ w ) −
1. Moreover, since ∆( w ) ∈ U m,p ,p k , there is a j ≤ h ∗ = h p − σ ( m ) p k − p i such that jp k + σ ( m ) < ∆( w ) < ( j + 1) p . Since p k > p k − , then jp k − + σ ( m ) < ∆( w ), with j ≤ h p − σ ( m ) p k − − p i . This means∆( w ) ∈ U m,p ,p k − as for Theorem 3.1.2. Finally, if ∆( w ) < p k all requirements of Theorem 3.1.2 are satisfied.Despite of the above remark, the conditions of Theorem 3.1.1. and 3.1.2 are not equivalent. For example, m = 2 , p = 11321, p = 12583 and p = 13093 verify conditions 3.1.1 (so w = mp p p is a primitive weirdnumber) but not 3.1.2, because p < ∆( w ) = 43936. In the other sense, for m = 2 , p = 23143, p = 24043, p = 27061 and p = 3077507, conditions 3.1.2 are satisfied, but the weird number ˜ w = mp p p p does notverify the conditions 3.1.1, because ∆( ˜ w ) = 39680 U m,p ,p .4. The application of Theorem 3.1
Theorem 3.1 may be used to develop an algorithm which searches for primitive weird numbers with manydifferent prime factors. The sufficient conditions are computationally much easier to check than the standarddefinition of weird number. The question, however, is in which range the prime numbers p , . . . , p k should bechosen. The following theorem gives a partial answer. Theorem 4.1.
Let m be a deficient number and let d be its deficience. Let p , . . . , p k be primes, with m < p
km/d − k/ then ˜ w is deficient.Proof. (i). We first prove that if p , p , . . . , p k are distinct primes with m < p < · · · < p k < km/d − ( k + 2) / w = mp . . . p k is abundant.Since ( m, p i ) = 1, one has σ ( w ) = σ ( m ) σ ( p ) . . . σ ( p k ). This implies that: σ ( w ) w = (cid:18) − dm (cid:19) k Y h =1 (cid:18) p h (cid:19) > (cid:18) − dm (cid:19) · (cid:18) p k (cid:19) k > . The above inequalities hold because for positive x , the function x → /x is decreasing, and because theequation (cid:18) − dm (cid:19) · (cid:18) q (cid:19) k = 2holds for q = 1 k s − dm − kd m − k + 12 + O (cid:18) dm (cid:19) > p k . Therefore w is abundant. (ii). The proof is analogous. (cid:3)
If one wants to generate weird numbers by means of Theorem 3.1 (3.1.1 or 3.1.2), i.e., abundant numbers ˜ w with∆( ˜ w ) ∈ U m,p ,p k on a hand U m,p ,p k have to be as large as possible. Good choices are with p − σ ( m ) as largeas possible. On the other hand, in order to get higher values of h ∗ , p k − p have to be as small as possible. Thisleads to consider k -tuples of primes p , . . . p k in an interval that, by Theorem 4.1, includes 2 km/d − ( k + 1) / k ≤ d then 2 km/d might be smaller than σ ( m ), and if p < σ ( m ) then h ∗ < U m,p ,p k isempty. Therefore, k > d is generally preferable, and all new weird numbers we have found enjoy this property.The primitive weird numbers generated with Theorem 1 in [8] are particular cases of Theorem 3.1.1, with m = 2 h , and k = 2. When p and p are chosen according to that theorem, then conditions of Theorem 3.1.1are fulfilled and w = 2 h p p is a primitive weird number. RIMITIVE WEIRD NUMBERS HAVING MORE THAN THREE DISTINCT PRIME FACTORS 5
However, Theorems 3.1.1 and 3.1.2 become more interesting when applied to search weird numbers with severalprime factors.Theorem 3.1.1 yields primitive weird numbers of the form mp . . . p k where m is a deficient number, k is aninteger larger than the deficience of m and the p i ’s are suitably chosen. It is relatively easy to generate primitiveweird numbers up to four distinct prime factors. The table below shows some of the primitive weird numbershaving at least five distinct prime factors we were able to generate with Theorem 3.1.1 w prime factorization m ∆( w )9210347984 2 · · · ·
523 2 · · · ·
541 2 ·
17 30423941578736 2 · · · · · · · · · · · ·
823 2 · · · · · · · · · · · · · · · · ·
257 7846432852586770937891968 2 · · · · ·
257 8518432892333375893455232 2 · · · · ·
257 7673633622208489084493184 2 · · · · ·
257 72832
Table 1.
The above primitive weird numbers have five distinct prime factors and have beenfound by means of Theorem 3.1.1 with k = 4 for m = 2 h and with k = 3 for m = 136 = 2 · m = 32896 = 2 · h p p p .By going deeper in the implementation of Theorem 3.1.2, we have been able to find 65 primitive weird numbers w having five distinct prime factors of the form w = 2 h p p p p ; and nine primitive weird numbers having sixdistinct prime factors, that are shown in the table below. As a comparison, before our computations only threeprimitive weird numbers having six distinct prime factors were known at the OEIS database [10]. w prime factorization ∆( w )44257207676 2 · · · · ·
881 8125258675788784 2 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Table 2.
The above primitive weird numbers have six distinct prime factors and have beenfound by means of Theorem 3.1.2 with k = 4 for m = 2 h and k = 3 for m = 32896 = 2 · Tracking weird numbers with several distinct prime factors and eventual odd weirdnumbers
As we have seen, Theorems 3.1.1 and 3.1.2 provide two distinct strategies to track primitive weird numbers withseveral distinct prime factors. The same approaches could be applied to track odd weird numbers. If m > w that one creates by means of Theorem 3.1.1 or 3.1.2 are odd, and if the conditions arefulfilled, w would be a primitive odd weird number.For tracking a weird number with several distinct prime factors (even or odd) both strategies (3.1.1 and 3.1.2)start with choosing a deficient number m with deficience d . As a general rule, since we want k > d , small valuesof d have to be preferred in order to keep the computational complexity low. Indeed, the case d = 1 correspondsto m = 2 h assuming that no further almost perfect numbers exist. This case has been largely discussed in[6, 8]. The only known composite numbers with d = 2 are even. For example, m = 136, m = 32896 havedeficience 2 [10, Sequence A191363] and, as shown in Table 1 and Table 2, Theorem 3.1 allows to find severalprimitive weird numbers starting with such values of m . We expect that primitive weird numbers could begenerated also from m = 2147516416, whose deficience is 2, both with Theorem 3.1.1 and 3.1.2. Unfortunatelythe computation becomes dramatically longer.As far as we know there are no known integers with deficience d = 3. The only known integers with deficience d = 4 are even, and the only known integer with d = 5 is 9 for which the above approaches are hard to apply. G. AMATO, M. F. HASLER, G. MELFI, AND M. PARTON
An interesting case is d = 6. There are several integers whose deficience is 6, some of which are odd. Thelist starts with 7, 15, 52, 315, 592, 1155, 2102272, 815634435, and no other terms are known [10, SequenceA141548].So, if one wants to track odd weird numbers, the starting m in both approaches 3.1.1 and 3.1.2, could be m = 315, m = 1155 or m = 815634435. Unfortunately our attempts to track an odd weird numbers from sucha choice of m have been unfruitful. 6. Conclusion
As one can argue from a table of primitive weird numbers, most of the primitive weird numbers are of the form2 k pq for k ∈ N , and primes p and q . This was already pointed out in [8] where the third author, among otherthings, conjectured that there are infinitely many primitive weird numbers of the form 2 k pq .It seems that primitive weird numbers that are not of this form become rarer. For example, between the 301stand the 400th, there are only 7 primitive weird numbers that are not of the form 2 k pq . Five of them have fourdistinct prime factors and two of them have five distinct prime factors. The existence of several weird numbershaving six distinct prime factors leads to the following conjecture. Conjecture 6.1.
Given an integer k ≥ , there exists a primitive weird number having at least k distinct primefactors. Of course, a positive answer would settle the question of the infiniteness of primitive weird numbers.However, a proof that primitive weird numbers have a bounded number of distinct prime factors would notsettle neither the question of the infiniteness of primitive weird numbers nor the question of the existence ofodd weird numbers.
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Elementary problem E2308 , Amer. Math. Monthly (1972), 774. Cited on page 1.[2] Benkoski, S.T. and Erd˝os, P., On weird and pseudoperfect numbers , Mathematics of Computation, (1974), 617–623. Citedon page 1.[3] Cram´er, H., On the order of magnitude of the difference between consecutive prime numbers , Acta Arithmetica (1936),23–46. Cited on page 1.[4] Friedman, C.N., Sums of divisors and Egyptian Fractions , Journal of Number Theory (1993), 328–339. Cited on page 1.[5] Guy, R.K., “Unsolved Problems in Number Theory”, Third Edition, Springer, 2004. Cited on page 1.[6] Iannucci, D.E., On primitive weird numbers of the form k pq , (2015), arXiv:1504.02761v1. Cited on pages 1 and 5.[7] Kravitz, S., A search for large weird numbers , Journal of Recreational Mathematics (1976), 82—85. Cited on page 1.[8] Melfi, G., On the conditional infiniteness of primitive weird numbers , Journal of Number Theory (2015), 508-514. Citedon pages 1, 2, 4, 5, and 6.[9] Pajunen, S.,
On primitive weird numbers , in “A Collection of manuscripts related to the Fibonacci sequence”, V.E. Hoggatt,Jr. and M. Bicknell-Johnson (Eds.) 1980, Fibonacci Association, 162–166. Cited on page 1.[10] Sloane, N.J.A.., “The On-line Encyclopedia of Integer Sequences”,
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Universit`a di Chieti-Pescara, Dipartimento di Economia Aziendale, viale della Pineta 4, I-65129 Pescara, Italy
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University of Applied Sciences of Western Switzerland, HEG Arc, Espace de l’Europe, 21, CH-2000 Neuchˆatel
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