DIVISOR ADVISORY AND OTHER TALES

The origin of this article can be traced to an article by Steven Kahan [1]. I was fascinated to see a new, albeit coincidental, property of numbers drop from the sky. Of course, this is the sort of thing that one might expect when one jostles billions of numbers around.

Kahan points out that if the prime factorization of a positive integer n is given by

p1a1 p2a2 p3a3 ... pkak

then the total number of divisors of n is given by the formula

t (n) = (a1+1)(a2+1)(a3+1) ...(ak+1)

The coincidental property which also gives the total number of divisors of n is given by the formula

t *(n) = p1a1 + p2a2 + p3a3 + ... + pkak

Of course, this coincidental property only applies to a relatively small percentage of n.

Kahan checked all values of n less than one million to determine when t (n) = t *(n). This resulted in a table showing all 142 occurrences for n less than one million. Within the table was one solitary odd example. Specifically

n = 440895 = 3*5*7*13*17*19 and t (n) = t *(n) = 64

Kahan asked why was there such a scarcity of odd examples. A brief examination of Kahan's table reveals the difference between the one odd example and all others. The exponents of the factors of the odd example do not have values greater than one, whereas all the other examples do, and the odd example does not have 2 as a prime factor; an obvious requirement for n to be odd. Note also that t (n) = t *(n) = 64 = 26.

If we require that a1 = a2 = ... = ak = 1, and apply the formula for calculating t (n) we see that

t (n) = (a1+1)(a2+1)...(ak+1) = (1+1)(1+1)...(1+1) = 2k

Since all of the ak = 1 we can contrive an odd value for n where t (n) = t *(n) by finding primes, other than two, who's sum is 2k,and k is even. For example

n = 261583125881 = 7*11*29*31*37*41*47*53

and t (n) = t *(n) = 256

Several other examples, all easily calculated by hand, are shown in Table 1.

Table 1.

n

Factors of n

t(n) = t*(n)

64957990097

7*11*13*17*23*29*59*97

256

14215288950725317

257*251*181*131*53*47*41*13*7

1024

6185899960201112210126330532129875492626589167485

11047*11027*11003*5009*5003*4013*4007*4003*4001*2003*1999*1801*601*11*5*3

65536

185000294736463679006360385

65003*71*61*59*53*47*41*37*31*29*23*17*13*5*3

65536


Even values for n can be obtained using 2 as one of the prime factors of n and applying the same ideas. Table 2 shows some examples.

Table 2.

n

Factors of n

t (n) = t*(n)

10357410

2*3*5*7*31*37*43

128

607845706230

2*3*5*7*41*53*71*73*257

512

79652448604326

2*13*31*37*41*43*109*113*123

512


In conclusion I believe, but have not proven, the following:

1. There is an infinite number of cases where t (n) = t *(n).

2. t (n) = t *(n) = 2k can be calculated for all 2k for k ³ 6.

3. If k is even, then n will be odd. If k is odd, then n will be even.

 

References:

1. Steven Kahan, Divisor Advisory, Journal of Recreational Mathematics, 30:1, pp. 41-44, 1999-2000