A few days ago, I added a new page dedicated to Armstrong numbers to my collection of information on digital invariants.
What is little recognized is that Armstrong defined four different types of Armstrong numbers, which he called Armstrong numbers of the first-, second-, third-, and fourth kind. Of special interest are Armstrong numbers of the third kind. Such a number, n, represented by digits dm, dm-1, …, d1, is equal to
I have not seen such a definition before by any name. An example of an Armstrong number of the third kind (in base-10) is 3435, since
3435 = 33 + 44 + 33 + 55 = 27 + 256 + 27 + 3125 = 3435
Update: Almost as soon as I posted this note, I discovered Munchausen numbers, defined by Daan van Berkel in the same way as Armstrong numbers of the third kind. In a brief 2009 paper, he exhibited, inter alia, all the Munchausen numbers, alias Armstrong numbers of the third kind, in bases 2 through 10. (There aren’t many.) Apparently, 3435 is the only nontrivial base-10 Munchausen number. (Note that this result relies on defining 00 to be 1.) Apparently such numbers are also called perfect digit-to-digit invariants, or PDDIs. (See Wikipedia article here.) I will be updating my page about Armstrong numbers to incorporate this additional information as soon as I can.