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. (So 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 is 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
Definition.
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. |
