A Smith number is a composite number, the sum of whose digits equals the
sum of the digits of all its prime factors.
The smallest Smith number is
4 = 2×2.
The sum of the digits of 4 is 4, and
The sum of digits of its prime factors is 2 + 2 = 4.
Another example is
6,036 = 2×2×3×503;
6,036 has a digit sum of 15, and
the sum of the digits of its prime factors is also 15.
How many Smith numbers are there between 2 and 10,000? ... In 1987
Wayne McDaniel showed that infinitely many Smith
numbers exist.
Patrick Costello & Kathy Lewis in Mathematics Magazine (June 2002);
"Lots of Smiths," Vol. 75, No. 3. (Jun., 2002), pp. 223-226.
Stable Link; Downloaded
Version
The first ten are
4, 22, 27, 58, 85, 94, 121, 166, 202, and 265.
The last three are
9,942; 9,975; and 9,985.
"Albert Wilansky noticed Dr. Harold Smith's phone number 493-7775 when
written as the single number 4937775 ... is a composite number where the sum of the digits
in its prime
factorization is equal to the digit sum of the number. Adding the digits
in the number
and the digits of its prime factors 3, 5, 5, and 65837 resulted in
identical sums of 42."
[ 4937775 = 3(5)(5)(65837)
4+9+3+7+7+7+5 = 42 = 3+5+5+6+5+8+3+7;
see "Lots of Smiths"]
Another such number found by him is 6036.
W. McDaniel, The existence of infinitely many k-Smith numbers, Fibonacci
Quarterly 25 (1987), 76-80.
Density
issues with Smith's could be explored [email Don Dechman for a QBasic computer program: dondechman@verizon.net]. A comparison with
other numbers' - e.g.,
Primes,Ulam numbers) -
asymptotic density may be of some interest.