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A natural number n is called perfect if σ(n) = 2n and multiply perfect whenever σ(n) is a multiple of n. In 1956, Erdős published improved upper bounds on the counting functions of the perfect and multiply perfect numbers [2]. These estimates were soon superseded by a theorem of Wirsing [15] (Theorem B below), but Erdős’s methods remain of interest as they are applicable to more general questions concerning the distribution of gcd(n, σ(n)). Erdős describes some applications of this type (op. cit.) but omits the proofs. In this paper we prove corrected versions of his results, and we establish some new results in the same direction. For real numbers x ≥ 1 and A ≥ 1, put G(x,A) := #{n ≤ x : gcd(n, σ(n)) > A}.

On the greatest common divisor of a number and its sum of divisors