3- digit Mania

Dear Calvin, You are correct. I posted these numbers called Narcissistic numbers as an oddity.

An n-digit number that is the sum of the nth powers of its digits is called an n-narcissistic number. It is also sometimes known as an Armstrong number, perfect digital invariant (Madachy 1979), or plus perfect number.

Hardy (1993) wrote, "There are just four numbers, after unity, which are the sums of the cubes of their digits: 153=1^3+5^3+3^3, 370=3^3+7^3+0^3, 371=3^3+7^3+1^3, and 407=4^3+0^3+7^3.

The smallest example of a narcissistic number other than the trivial 1-digit numbers is 153=1^3+5^3+3^3.

The first few are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ...

It can easily be shown that base-10 n-narcissistic numbers can exist only for n<=60, since n·9^n<10^(n-1) for n>60.

In fact, a total of 88 narcissistic numbers exist in base 10, as proved by D. Winter in 1985 and verified by D. Hoey. T. A. Mendes Oliveira e Silva gave the full sequence in a posting to sci.math on May 9, 1994.

As a matter of fact Hardy (1993) wrote, "There are just four numbers, after unity, which are the sums of the cubes of their digits: 153=1^3+5^3+3^3, 370=3^3+7^3+0^3, 371=3^3+7^3+1^3, and 407=4^3+0^3+7^3. These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician."

If you agree with the above statement, my apologies for posting this oddity. Please delete the problem. :) Regards Satyen

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