New amicable four-cycles

Fifty new amicable four-cycles are discovered by the constructive method invented in 1969 by the second author. 1. Let τ(n) denote the sum of proper divisors of a natural number n, and let σ(n) = n + τ(n). We consider when the sequence n, τ(n), τ (n) := τ(τ(n)), . . . becomes periodic. If n = τ (n) with k minimal, then n1 = n, n2 = τ(n), n3 = τ (n), . . . , nk = τ (k−1)(n) is called an amicable k-cycle. The study of amicable 1-cycles (perfect numbers) and of amicable 2-cycles (amicable pairs) has a thousandyear-old history. Here we study amicable four-cycles. The smallest example is n1 = 2 · 5 · 17 · 3719, n3 = 2 · 521 · 829, n2 = 2 · 5 · 193 · 401, n4 = 2 · 40787, discovered by H. Cohen [5] in 1970 by an exhaustive trial and error search below 60, 000, 000. 2. Alternatively, one may try to construct amicable four-cycles of a special form. This can be done by means of the following theorem, due to the second author. Theorem 1 ([3]). Let a1 and a2 be natural numbers, a1 6= a2, and let D := a1 a2− τ(a1) τ(a2). Let d1 d2 = a1 a2 be any factorization into two natural numbers d1, d2. Consider the six numbers pij, ri (i, j = 1, 2) pij := 1 D (τ(ai+1)σ(ai) + dj σ(ai+1)), where a3 := a1, (1) ri := 1 ai τ(ai pi1 pi2). (2) If all six are primes, and pij | 6 ai, ri | 6 ai, pi1 6= pi2 (i, j = 1, 2), then the following is an amicable four-cycle: n1 = a1 p11 p12, n3 = a2 p21 p22, n2 = a1 r1, n4 = a2 r2 . The smallest example is n1 = 3 · 5 · 7 · 83 · 359, n3 = 3 · 5 · 11 · 79 · 263, n2 = 3 · 5 · 7 · 31643, n4 = 3 · 5 · 11 · 20183, which was found in [3] in 1969 without use of a computer. Received by the editor September 19, 2001 and, in revised form, February 18, 2002. 2000 Mathematics Subject Classification. Primary 11A25.