Definition:Smith Number

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Definition

A Smith number is a composite number for which the sum of its digits is equal to the sum of the digits in its prime decomposition.


Sequence of Smith Numbers

The sequence of Smith numbers begins:

$4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, \ldots$


Also known as

Smith numbers are also referred to by some sources as joke numbers.


Also see


Historical Note

The term Smith number was coined by Albert Wilansky.

He noticed the property in the phone number ($493$ - $7775$) of his brother-in-law Harold Smith:

$4937775 = 3 × 5 × 5 × 65837$

while:

$4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42$


The full text of the article in which this appeared:

A Smith number is a composite number the sum of whose digits is the sum of all the digits of its prime factors. The (rather startling) reason for the name is mentioned below.
Examples. $9985 = 5 \times 1997$, $9 + 9 + 8 + 5 = 5 + 1 + 9 + 9 + 7$, $6036 = 2 \times 2 \times 3 \times 503$, $6 + 0 + 3 + 6 = 2 + 2 + 3 + 5 + 0 + 3$.
The number of Smith numbers between $n$ thousand and $n$ thousand $+ 999$ for $n = 0, 1, 2, \ldots, 9$, is, respectively, $47$, $32$, $42$, $28$, $33$, $32$, $32$, $37$, $37$, $40$.
I wonder whether there are infinitely many Smith numbers.
The largest Smith number known is due to my brother-in-law H. Smith who is not a mathematician. It is his telephone number: $4937775$!


Sources