82.27 An interesting number fact

Abstract

Next we give a couple of examples to illustrate the procedure given above. Example I: Let S = 7 263 025 = 5 2 x 7 4 x l l 2. Here N = 2 4 x 3 2 x 5 2 = 3600. Note that d(S) = d(N) = 5 x 3 x 3 = 45. We can take M = 2 3 x 3 2 x 5 x 7 = 2520 and d(M) = 4 x 3 x 2 x 2 = 48. Note that M < N < S and while each of S and N has 45 divisors M has 48 divisors. Example II: Let S = 148 225 = 5 2 x 7 2 x 11 2. Here N = 2 2 x 3 2 x 5 2 = 900 and N < S. We can take M = 2 4 x 3 2 x 5 = 720. Now M < N < S. Note that d(M) = 5 x 3 x 2 = 30 and d(S) = d(N) = 3 x 3 x 3 = 27. So M has more divisors than N (and so than S). I thank the referee for his suggestions.

Generalising ‘Sums of cubes equal to squares of sums’

“…JOHN MASON David Pagni [1] drew attention to a result which is ascribed by Dickson [2, p. 286] to Liouville (1857) [3], that the sum of the cubes of the number of divisors of each of the divisors of an integer, is equal to the square of their sum. For example, the divisors of 6 are 1, 2, 3, and 6, which have 1, 2, 2, and 4 divisors respectively, and l 3 + 2 3 + 2 3 + 4 3 = 81 = (1 + 2 + 2 + 4) 3 .…”