I've often heard without attribution the 'proof' that all natural numbers are interesting (because being the smallest uninteresting number is an interesting property). Along those lines, I've decided to collect lists of numbers with the interesting property of being deemed uninteresting.
Ramanujan's taxicab number 1729 = 12³ + 1³ = 10³ + 9³ is perhaps the most famous interesting number. In order to list these numbers in a concrete way, I have found several lists of interesting numbers
and given the first few positive integers missing from those lists. These are of course subject to change as interesting properties are discovered; these lists are as of September 2006.
The Penguin Dictionary of Curious and Interesting Numbers*: 43, 51, 54, 57, 58, 62, 67, 68, 74, 75, 78, 80, 82, 83, 86, 87, 92, 93, 95, ...
The Penguin Dictionary of Curious and Interesting Numbers**: 54, 57, 58, 67, 75, 78, 80, 82, 83, 92, 93, 95, 96, 106, 107, 109, 115, ...
What's Special About This Number?: 226, 234, 258, 277, 295, 298, 304, 316, 329, 332, 338, 339, 349, 356, 361, 363, 380, ...
Prime Curios!†: 338, 392, 395, 398, 402, 404, 405, 406, 408, 412, 413, 415, 416, 417, 418, 422, 424, 425, 426, 428, ...
What's Special About This Number?: 304, 305, 306, 309, 310, 314, 316, 321, 324, 326, 328, 329, 331, 332, 334, 338, 339, 346, ...
Notable Properties of Specific Numbers: 22, 36, 38, 46, 48, 49, 50, 51, 53, 54, 57, 58, 59, 62, 63, 67, 68, 71, 73, 74, 76, ...
Properties of the First 5000 Integers: 33, 50, 69, 74, 77, 98, 104, 132, 134, 135, 138, 140, 143, 146, 150, 152, 154, 155, 156, 159, ...
Online Encyclopedia of Integer Sequences‡: 8795, 9734, 9935, 10418, 10776, 10895, 11147, 11446, 11455, 11612, 11630, 11641, 11694, 11722, ...
I will note that 338 is uninteresting in the first six lists above, making it a good candidate for an all-around uninteresting number. Beyond that, the numbers listed for the OEIS do not appear in the other lists; this seems to be the ultimate test for interesting numbers.
Of 39: This appears to be the first uninteresting number, which of course makes it an especially interesting number...
(David Wells, in his first edition)
Of 51: This appears to be the first uninteresting number, which of course makes it an especially interesting number...
(David Wells, in his second edition)
I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.
(G. H. Hardy)No,
he replied, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.
* Original edition, 1986.
** Revised edition, 1997.
† After I wrote this page, I discovered that Chris Caldwell has a similar page about the numbers he was missing from his curios list. His list does not match mine; at the time of writing, it did not include 338 or 392, neither of which has an entry.
‡ This database is of course not designed as a list of interesting numbers, but arguably interesting numbers should appear in some sequence. Since all positive numbers are in some list, extended appropriately (A000027 and in either A018252 or A000040, etc.) this list includes only those enumerated.