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The numbers we are interested in are sometimes classified as narcissistic numbers. A narcissistic number is usually described as one whose value can be expressed as some function of its individual digits. This is not a totally satisfactory definition, as it leaves the nature of the function unconstrained and does not exclude the obvious (and therefore the uninteresting). For example, 125 = 1∑102 + 2∑101 + 5∑100. If we declare 125 to be narcissistic on the basis of this observation, then any rational number is narcissistic. Of course, the term is usually reserved for more remarkable facts, that 153 = 13 + 53 + 33, for instance.

Because the properties we want to describe depend upon the representation of numbers in standard positional number systems, we begin by formalizing that representation. In what follows, all numbers are assumed to be non-negative integers.

Numbers are written in base-b (or radix-b) notation using b distinct symbols D0, D1, ..., Db-1, that represent, respectively, 0, 1, ..., b-1. If d' is a variable representing one of these symbols, then we will use d to designate the corresponding value. Clearly distinguishing numerals from numbers is admittedly tedious, but it technically is required, even though it is seldom done. We will, at times, be less obsessive about this distinction when no ambiguity results.

Definition. If n is a number with representation d'md'm-1 ... d'1 in base-b notation, then

where dm is not 0, except possibly if m = 1. We call the value n the positional value in base b of representation d'md'm-1 ... d'1, written Pb(d'm ... d'1). The order-k summation value of representation d'md'm-1 ... d'1, written Sk(d'm ... d'1) is the sum

Let n be a number with base-b representation R. We say that n is a perfect digital invariant (or PDI) of order k in base b if n = Pb(R) = Sk(R). More simply, a number is a PDI of order k in base b if its value is obtained when its digits in the base-b representation are raised to the kth power and added together. Thus, 4151 is an order-5 decimal (i.e., base-10) PDI, since 4151 = 45 + 15 + 55 + 15.

Definition. A pluperfect digital invariant (or PPDI) in base b is a base-b PDI of order m, where m is the number of digits in its base-b representation. In other words, a PPDI is a PDI for which its digits are raised to the power equal to the number of digits in the representation. The number 153 is a PPDI in base 10, since 153 = 13 + 53 + 33.

Definition. A hyperperfect digital invariant (or HPDI) is a base-b PPDI whose length, in base-b notation, is b. Because 17 = 1223 = 13 + 23 + 23, it is a base-3 HPDI.


N.B. Most discussions of PDIs, PPDIs, and HPDIs limit concern to positive integers. As noted above, the discussion here applies to non-negative integers. There seems to be no compelling reason to exclude considering zero. Thus, whereas most tabulation of base-10 PPDIs begin with 1, the list here begins with 0, since 0 = 01.


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