A Smith number
is a composite numbers
such that sum of digits of
is equal to the sum of digits of the prime factors of
, counted with multiplicity. They are also called joke numbers
For example, 666 is a Smith number since and
Note that hoax numbers are defined
similarly, but they do not take into account prime factors multiplicity.
Clearly, a squarefree hoax number is also a Smith number.
The first pair, triple, etc. of consecutive Smith numbers (also
called Smith brothers)
728, 73615, 4463535, 15966114, 2050918644, 164736913905,
and 8090674745553, respectively.
If the requisite of compositeness is left aside, then all the prime numbers trivially satisfy the condition on the sum of digits and it is possible to find a run of length 9 starting at 1979414080360.
Below, the spiral pattern of Smith numbers up to . See the page on prime numbers for an explanation and links to similar pictures.
The smallest 3 × 3 magic square made of Smith numbers is
The first Smith numbers are
4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438 more terms
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many Smith numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p