Νοτε Ον Consecutive Abundant Numbers

A positive integer N is called an abundant number if σ(N) > 2N, where σ (N) denotes the suns of the divisors of N including 1 and N. Abundant numbers have been recently investigated by Behrend, Chowla, Davenport , myself, and others ; it has been proved, for example, that they have a density greater than 0 . I prove now the following THEOREM. We can find two constants c l , c 2 such that , for all sufficiently large n, there exist cl log log log n consecutive integers all abundant and less than n , but not c 2 log log log n consecutive integers all abundant and less than n .