Essays/Unforgettable Numbers

In August 2009, Kip Murray asked for the "unforgettable" times seen on a 24-hour digital clock.

The problem is that every number has something notable about it, so that each number is "unforgettable"
and consequently it's hard to remember any single one of them.

0000	all zeros
0001	first counting number
0002	first prime number
0003	first odd prime
0004	first composite number
...

   24 60 #: ?. */ 24 60
1 6

0106	first number greater than 100 with 2 prime factors

etc.

You may have heard of the following story about Hardy and Ramanujan. One day Hardy took a taxi to visit Ramanujan. On arriving Hardy told Ramanujan that the taxi had the thoroughly unremarkable 4-digit number n on its license plate. Ramanujan immediately remarked that n is the first number that ... . I forget what n or the property was, something like, n is the first number that can be written as the sum of two perfect cubes in two different ways, something typically Ramanujanian.

Yes, that was it:

   c=: i*i*i=: >:i.200     NB. cubes of numbers from 1 to 200
   t=: (</~i.200) * +/~c   NB. upper triangle of the addition table of these cubes
   d=: </.~ ,t             NB. group the unique sums
   (2=#&>d)#d              NB. groups having exactly 2 members
┌─────────┬───────────────┬───────────────┬─────────┬─
│1729 1729│1092728 1092728│3375001 3375001│4104 4104│...
└─────────┴───────────────┴───────────────┴─────────┴─
   <./ {.&> (2=#&>d)#d     NB. the smallest number in these groups
1729
   I. , 1729 = t           NB. indices in the addition table where 1729 occurs
11 1609
   1 + ($t) #: 11 1609     NB. original numbers corresponding to the indices
1 12
9 10
   +/ 1 12 ^ 3             NB. check the sum of these numbers cubed
1729
   +/ 9 10 ^ 3
1729

Now that I have worked out the number I can find the story on the net. On hearing the story, J.E. Littlewood remarked that "every positive integer is one of Ramanujan's personal friends".

p.s. In my youth, when I needed to remember a (5-digit) number for a time, I would try to compute its largest prime factor by mental calculation. Try it and you'll see why it works.

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Contributed by Roger Hui.