Narcissistic Numbers in Various Bases
The sequence of "base 10" narcissistic numbers starts: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474 ... (sequence A005188 in OEIS)
The sequence of "base 3" narcissistic numbers starts: 0, 1, 2, 12, 22, 122
The sequence of "base 4" narcissistic numbers starts: 0, 1, 2, 3, 130, 131, 203, 223, 313
The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the kth powers of a k digit number in base b is
and if k is large enough then
in which case no base b narcissistic number can have k or more digits. Setting b equal to 10 shows that the largest narcissistic number in base 10 must be less than 1060.
There are only 88 narcissistic numbers in base 10, of which the largest is
- 115,132,219,018,763,992,565,095,597,973,971,522,401
with 39 digits.
Unlike narcissistic numbers, no upper bound can be determined for the size of PDIs in a given base, and it is not currently known whether or not the number of PDIs for an arbitrary base is finite or infinite.
Read more about this topic: Narcissistic Number
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