— Burt Kaliski Jr. (@modulomathy) June 29, 2012
The “profile” of a number is a term I made up for this problem (though others may have used it first). It seemed like a good way to describe how a number looks in terms of the sum of its prime factors (including repeats).
In general, larger numbers will have more and/or larger prime factors, so their profile will be larger. The problem is asking for an example that goes in the opposite direction: A number larger than 180 with a profile smaller than 15.
Profiles are to numbers like perimeters are to area or volume. A 3×3 square, for instance, has a perimeter of 3+3+3+3 = 12, and an area of 3*3 = 9. In terms of numbers, 9, which has a factorization of 3*3, has a profile of 3+3 = 6 — half the perimeter because a side in each dimensions is only counted once. For a given number of dimensions or factors including repeats, however, the ratio between perimeter and profile remains the same. Therefore, we’re interested in finding the polygon “shape” that gives the greatest perimeter for a given area or volume, in a given dimension. That shape, as it turns out, is a regular polygon or its higher dimensional equivalent. A square has the smallest perimeter for a given volume of all two-dimensional polygons; a cube, of all three-dimensional polygons, and so on. It follows that a number that is the power of a single prime has the smallest profile.
As a first test whether there is indeed a number larger than 180 with a profile smaller than 15, we can just look at the minimum possible profiles for each dimension or number of prime factors (including repeats), for numbers starting at 181. If the number of prime factors is d, then the smallest possible profile will be d*1811/d. Note that this doesn’t mean that such a factorization is actually possible — indeed, 1811/d isn’t even prime for any d > 1. What it means is that any other factorization will have a larger profile, just as any other shape will have a larger perimeter. For larger numbers than 182, the smallest possible profile based on this analysis will be even larger.
The smallest possible profiles for different values of d are as follows:
|Dimensions (d)||Minimum profile(d*1811/d)|
The minimum profiles continue to increase from that point as the number of dimensions increase.
From a practical perspective, the table can be understood as saying that the minimum profile if there’s only one prime factor (including repeats) is 181, which of course is correct because 181 is prime. If there are two prime factors, the minimum is just under 27. Although 181 is prime, it is clear that if it had two prime factors, they would have to add up to more than 26, because the product of 13 and 13 is 169.
Because the actual profile must be an integer, the actual minimum possible for any number of dimensions is 15. This means that the answer to the question is no: there is no number larger than 180 with a smaller profile than 15.