POWERFUL AMICABLE NUMBERS PAUL POLLACK P Abstract. Let s(n) := djn; d<n d denote the sum of the proper di- visors of the natural number n. Two distinct positive integers n and m are said to form an amicable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first example of an amicable pair, known already to the ancients, is f220; 284g. We do not know if there are infinitely many amicable pairs. In the opposite direction, Erd}osshowed in 1955 that the set of amicable numbers has asymptotic density zero. Let ` ≥ 1. A natural number n is said to be `-full (or `-powerful) if p` divides n whenever the prime p divides n. As shown by Erd}osand 1=` Szekeres in 1935, the number of `-full n ≤ x is asymptotically c`x , as x ! 1. Here c` is a positive constant depending on `. We show that for each fixed `, the set of amicable `-full numbers has relative density zero within the set of `-full numbers. 1. Introduction P Let s(n) := djn; d<n d be the sum of the proper divisors of the natural number n. Two distinct natural numbers n and m are said to form an ami- cable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first amicable pair, 220 and 284, was known already to the Pythagorean school. Despite their long history, we still know very little about such pairs. Thus, we know over 10 million examples [19] (cf. [11]), but we have no proof that there are infinitely many. In the opposite direction, Erd}os[5] showed in 1955 that the set of amicable numbers has as- ymptotic density zero. The strongest result of this type is due to Pomerance [23]. Another unsolved problem concerns the existence of an amicable pair of opposite parity. It is easy to prove (see, e.g., [12]) that in such a pair, the odd member is a square and the even member is either a square or twice a square. Thus, it is natural to ask how often s(n2) is a perfect square or twice a square. Iannucci and Luca [16] have shown that the number of n ≤ x for which s(n2) is a square is bounded by x=(log x)3=2+o(1), as x ! 1, and their 2010 Mathematics Subject Classification. Primary: 11A25, Secondary: 11N37. Key words and phrases. amicable number, powerful number, sociable number, aliquot sequence. 1 2 P. POLLACK method yields the same estimate for how often s(n2) is twice a square. As a corollary, the set of n for which n2 is amicable has asymptotic density zero. Squares are an example of 2-full numbers. Here the natural number n is called `-full if every prime dividing n appears to at least the `th power in the prime factorization of n. It is known (see [9]) that the number of `-full 1=` numbers n ≤ x is asymptotically c`x , as x ! 1, where c` is a positive constant depending on `. Our main result is the following: Theorem 1.1. Fix ` ≥ 1. As x ! 1, only o(x1=`) of the `-full natural numbers not exceeding x are amicable. Note that the case ` = 1 is exactly the theorem of Erd}osquoted in the opening paragraph. Our method is entirely different from that of Iannucci and Luca. We adopt two strategies, depending on the parity of `. If ` is odd, we show that an `-full number n usually satisfies either n < s(n) < s(s(n)) or n > s(n) > s(s(n)). In neither case can n be amicable. The ideas here trace back to Erd}os[5] (see also [7]) and Erd}os{Granville{Pomerance{Spiro [8]. Suppose now that ` is even. We apply an elementary sieving argument to show that for most `-full n, the number s(n) has many primes appearing to the first power. Thus, σ(s(n)) is divisible by a large power of 2. On the other hand, if n is amicable, then σ(s(n)) = σ(n). For most `-full numbers n, nearly all of the primes dividing n show up to precisely the `th power; since ` is even, these prime powers have odd σ-value. Thus, for an `-full number n, we expect that σ(n) is only divisible by a small power of 2. So most `-full numbers are not amicable. Amicable numbers are a special class of sociable numbers. In the final section of this paper, we review what it means for a number to be sociable, and we prove the analogue of Theorem 1.1 for sociable numbers of a fixed odd order. Notation. Throughout, the letters p and q are reserved for primes. We write P (n) for the largest prime divisor of n, with the understanding that P P (1) = 1. We say that n is y-smooth if P (n) ≤ y. We let !(n) := pjn 1 stand for the number of distinct prime divisors of n, and we write rad(n) := Q pjn p for the largest squarefree divisor of n. If d divides n and gcd(d; n=d) = 1, we say that d exactly divides n, and we write d k n. We write νp(n) for the nonnegative integer satisfying pνp(n) k n (the p-adic order of n). We use POWERFUL AMICABLE NUMBERS 3 1C for the indicator function of the set defined by the condition C; so, for 2 example, 1v squarefree = µ (v). If S1 and S2 are subsets of the natural numbers, by the relative density of S1 within S2, we mean the limit #fn ≤ x : n 2 S1g lim ; x!1 #fn ≤ x : n 2 S2g if it exists. The upper and lower relative densities are defined analogously, with lim sup and lim inf replacing lim. We say that a property holds for almost all m if it holds for all m outside of a set of density zero (or relative density zero, depending on context). We use both the Landau{Bachmann O-notation and Vinogradov's notation; implied constants are absolute unless specified otherwise by subscripts. We write log1 x = maxf1; log xg, and for k > 1, we let logk denote the kth iterate of log1. 2. Preliminaries for the case of odd ` Fix an odd natural number `. We begin by proving the following theorem, the lower-bound half of which generalizes a theorem from [5] (see also [7]) and the upper-bound half of which generalizes [8, Theorem 5.1]. In both our notation and our general strategy, we follow [8] closely. Theorem 2.1. For each fixed > 0, all but o(x1=`) of the `-full numbers n ≤ x satisfy s(n) s(s(n)) s(n) (1) − ≤ ≤ + . n s(n) n Rather than treating all `-full numbers on an equal footing, we introduce a certain convenient partition: For each `-full number n, we write n in the form dv`, where d is (`+1)-full and v is squarefree. Then most `-full numbers correspond to small values of d. Indeed, the number of `-full n ≤ x for which d > y (say) is bounded by X X 1=` X −1=` 1=` − 1 1 ≤ x d ` x y `(`+1) : d>y v≤(x=d)1=` d>y d (` + 1)-full d (` + 1)-full It follows that to prove a result about almost all `-full numbers, such as Theorem 1.1 or Theorem 2.1, we may assume that the (` + 1)-full part d of n is fixed. Our task then becomes showing that as x ! 1, all but o(x) of the squarefree v ≤ x coprime to d are such that n = dv` satisfies the statement in question. 4 P. POLLACK For the rest of this section, we assume the (` + 1)-full number d is fixed. Whenever we use v below, we always mean a squarefree integer coprime to d, whether this is stated explicitly or not. Lemma 2.2. Fix a natural number T . For each value of v with 1 < v ≤ x, write ` ` dv = n1n2 and s(dv ) = N1N2; where P (n1N1) ≤ T and every prime dividing n2N2 exceeds T . Then, except for o(x) (as x ! 1) choices of v, we have n1 = N1. Proof. At the cost of excluding o(x) values of v ≤ x, we may assume that !−1 1=2 Y n1 ≤ d(log2 x) p : p≤T ` Indeed, in the opposite case, v has a T -smooth divisor e := n1=(n1; d) −1 1=2 Q exceeding (log2 x) p≤T p . For each such e, let e1 be the minimal ` ` 1=` integer with e j e1. Since e j v , we have that e1 j v; moreover, e1 ≥ e . So the number of v that may arise is at most X X X 1 (2) 1 ≤ x : e1=` e T -smooth v≤x e T -smooth 1=2 Q −1 1=2 Q −1 e>(log2 x) ( p≤T p) e1jv e>(log2 x) ( p≤T p) Observing that X 1 Y 1 1 = 1 + + + ::: < 1; e1=` p1=` p2=` e T -smooth p≤T we deduce that the right-hand side of (2) is o(x), as x ! 1. Now we show that for all but o(x) values of v ≤ x, the number σ(dv`) 2=3 is divisible by every natural number not exceeding (log2 x) . Let m be a natural number. For all but x=(log x)1='(m) exceptional values of v ≤ x, one can find a prime p ≡ −1 (mod m) for which p k v (see [22, Theorem 2]).