An Armstrong number is a number that is the sum of its own digits each raised to the power of the number of digits. This definition depends on the base b of the number system used, e.g., b = 10 for the decimal system or b = 2 for the binary system. It was named after Michael F. Armstrong. They are also called as narcissistic number , pluperfect digital invariant or a plus perfect number.In easy words we can say, An n -digit number equal to the sum of the nth powers of its digits. As:-
abcd... = pow(a,n) + pow(b,n) + pow(c,n) + pow(d,n) + ....
Eg. 153 = 13 + 53 + 33
The definition of a Armstrong number relies on the decimal representation n = dkdk-1...d1 of a natural number n, i.e.,
n = dk·10k-1 + dk-1·10k-2 + ... + d2·10 + d1, with k digits di satisfying 0 ≤ di ≤ 9. Such a number n is called Armstrong if it satisfies the condition
n = dkk + dk-1k + ... + d2k + d1k.
- The sequence of base 10 Armstrong numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, ...
- The sequence of base 8 Armstrong numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661,..
- The sequence of base 12 Armstrong numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 25, ᘔ5, 577, 668, ᘔ83,..
- The sequence of base 16 Armstrong numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 156, 173, 208, 248, 285, 4A5, 5B0, 5B1, 60B, 64B, ...
- The sequence of base 3 Armstrong numbers starts: 0, 1, 2, 12, 22, 122The sequence of base 4 Armstrong numbers starts: 0, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303
- In base 2, the only Armstrong numbers are 0 and 1.
In 1993 Hardy wrote, "There are just four numbers, after unity, which are the sums of the cubes of their digits. 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."
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