In the Groove

Sometimes you get into a groove, habits form and you don't go beyond the prescribed limits that you yourself have set. For some years, I've been using the Online Encyclopaedia of Integer Sequences (OEIS) when examining the mathematical significance of the number associated with my diurnal age. Now that my numbered days are approaching 26,000, I long ago realised that I was missing out on some important sequences because the number wasn't listed in the OEIS display.

25406 is a good example of what I mean. This number is a member of OEIS A025414:

a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways

3, 27, 54, 129, 194, 209, 341, 374, 614, 594, 854, 1106, 1314, 1154, 1286, 1746, 1634, 1881, 2141, 2246, 2609, 2889, 3461, 3366, 3449, 3506, 4241, 4289, 5066, 4826, 5381, 5606, 6569, 5561, 6254, 7601, 8186, 8069, 8714, 8126, 9434, 8921, 8774, 11066, 11574

Clearly 25406 is well short of making an appearance in the above list. However, it is listed in a Table of n, a(n) for n = 1..1000 where it is the 72nd entrant and thus the smallest number that can be represented as a sum of two nonzero squares in 72 different ways. But how to discover this when a search for 25406 using the OEIS search bar does not bring up this sequence?

The secret was to click on Hints and then reading through the contents of this page, one discovers the following little pearl of wisdom:

To search for a single large number in the OEIS, try Google, because Google has searched all the .txt files in the OEIS, and so may do a better job than the OEIS search engine.

Sure enough, this approach throws up a whole lot of other OEIS results that don't appear in the normal search, among them being OEIS A025414. This is a significant discovery because it increases the likelihood of finding an interesting sequence for large numbers such as 25406. The pedagogical principle at play here is to poke around and don't stay stuck within the same comfortable but limited perimeter. This is what had happened to me with the OEIS searching.