"Find a three-digit number that is the sum of the cubes of its digits." The problem was thus posed in the early 1960s on the pages of Litton Industries' Problematical Recreations series (reprinted in Angela Dunn's 1980 Mathematical Bafflers). They had originally published solutions of only 153 and 371 but some readers noted that 370 and 407 worked just as well.

The first time these four numbers may have been mentioned together in this sum-of-the-cubes-of-their-digits context is in a 1937 issue of Sphinx. H.S.M. Coxeter added the observation to his 1939 revision of W.W. Rouse Ball's Mathematical Recreations & Essays (11th edition), which is where G.H. Hardy found it for his 1940 A Mathematician's Apology.

By the early 1960s, the concept had been generalized: "Those integers (>1) that are equal to the sum of the nth powers of their digits are called perfect digital invariants (PDI's). The 4 numbers mentioned by Hardy are PDI's of the third order, that is, n = 3." [Joseph S. Madachy: 1966, 1979]

Harry L. Nelson had computed the tenth-order 4679307774 in 1963. By 1981, all 89 PDIs where the order was equal to the length of the number (called pluperfect digital invariants or PPDIs — the largest, 39 digits) had been tabulated. Work on other PDIs however seemed to languish. Lionel E. Deimel Jr. and Michael T. Jones saw fit to report their 41-digit discovery 36428594490313158783584452532870892261556 (order 42) in 1982 but as recently as 2004, internet-savvy recreational mathematicians were left to discover — or rediscover, as the case may be — the 15-digit 233411150132317 (order 17). It had been reported that extensions to 10^50 by G.N. Gusev and to 10^74 by Xiaoqing Tang had been achieved but actual numbers or their location failed to enter Web-consciousness.

Yesterday, all that changed: Joseph S. Myers computed and emailed a list of 255 PDIs < 10^105 to the Sequence Fanatics discussion list. I've formatted his results into this table, where the PDIs are followed by the order (or power) expressed as the length of the number plus (and in two instances, minus) a variance. The order is asterisked for the first two PDIs (0 & 1) because any positive power will work for them.