Recently (June 1999) an individual, by the name of Smith, sent in the following question to the MATH FORUM of Swarthmore College, in particular to the “**Ask Dr. Math**” feature of that website:

“If p(*n*) is the function which adds up all the divisors of a natural number *n*, then could you please list for me some numbers *n* which have the following properties:

- p(
*n*) has exactly one factor of 2; - p(
*n*)/2 has fewer distinct prime divisors than*n*itself.”

That person was answered by “Doctor Pete”, who said:

Hi,

Using Mathematica, for all *n* < 10000,

10 34 52 90 106 306 388 468 490 810 850 954 976 1300 1525 1666 1690 2650 2754 2890 3033 3050 3492 3610 3626 4194 4212 4410 5194 5200 5746 6066 6100 6292 6516 6800 7290 7650 8586 8746 8784 9610 9700

all satisfy both properties mentioned. Furthermore, these are the only such numbers in the range mentioned that satisfy both properties.For instance,

p(490) = 1026 = 2 × 3^{3} × 19

whereas

490 = 2 × 5 × 7^{2}.

I don’t know of any formula for finding such numbers; there doesn’t appear to be a pattern in the above list. But who knows, there may be an expression for them. To be honest I haven’t really looked into them very deeply.

– Doctor Pete, The Math Forum

It seems rather interesting to WTM that, given such a rather simple pair of conditions, there should be so few numbers that meet them. And there is that one “black sheep” (**3033**), a true “odd-ball”, a “rose among the thorns” (or vice versa). Numbers can be very, well, strange sometimes, can’t they?