Smith’s Problem

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:

  1. p(n) has exactly one factor of 2;
  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 × 33 × 19

whereas

490 = 2 × 5 × 72.

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?

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