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Let a1 )) > n. The following are proved: (1) If a, =p, a prime, for some I and p # (ai + aj)/2, 1 )) n for at most finitely many n. (b) If IPn I n. This conjecture has attracted many investigators and has been verified in the following special cases: (a) ai is square-free for all i (Marica and Schonheim [2]). (b) a, is prime (Winterle [3]). (c) n is prime (Szemeredi [4]). (d) n 1 is prime (Velez [4]). Velez [4] also considers another conjecture of Graham as to the form of those sequences for which equality is achieved and shows that this second conjecture implies the title conjecture. Here we prove two generalizations of Winterle's result (b). Part I. THEOREM 1. If A is the sequence al n. 1,j Received by the editors January 19, 1976. AMS (MOS) subject classifications (1970). Primary IOA05. ?D American Mathematical Society 1977

On a conjecture of Graham concerning greatest common divisors