International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 8, Pages 495-499

Graceful numbers

Kiran R. Bhutani and Alexander B. Levin

Department of Mathematics, The Catholic University of America, Washington, DC 20064, USA

Received 14 May 2001

Copyright © 2002 Kiran R. Bhutani and Alexander B. Levin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We construct a labeled graph D(n) that reflects the structure of divisors of a given natural number n. We define the concept of graceful numbers in terms of this associated graph and find the general form of such a number. As a consequence, we determine which graceful numbers are perfect.