Practical Number

A number $n$ is practical if for all $k\leq n$, $k$ is the sum of distinct proper divisors of $n$. Defined in 1948 by A. K. Srinivasen. All even Perfect Numbers are practical. The number

$$m=2^{n-1}(2^n-1)$$

is practical for all $n=2$, 3, .... The first few practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, ... (Sloane's A005153). G. Melfi has computed twins, triplets, and 5-tuples of practical numbers. The first few 5-tuples are 12, 18, 30, 198, 306, 462, 1482, 2550, 4422, ....

References

Melfi, G. ``On Two Conjectures About Practical Numbers.'' J. Number Th. 56, 205-210, 1996.

Melfi, G. ``Practical Numbers.'' http://www.dm.unipi.it/gauss-pages/melfi/public_html/pratica.html.

Sloane, N. J. A. Sequence A005153/M0991 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.