Smith number

A Smith number is a composite number, in which the sum of its digits is equal to the sum of all digits of its prime factors. The prime factors are given here without exponents and repeated according to the product presentation as often as necessary. ( 378 = 2 × 3 × 3 × 3 × 7 instead of 378 = 2 × 33 × 7 )

Example

The numeric sum of the number 166 is 1 6 6 = 13

166 = 2 × 83, the sum of the digits of its prime factors is thus 2 8 3, which also yields 13.

So 166 is a Smith number.

Smith numbers in the decimal

The first Smith - numbers in the decimal system are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378 ( sequence A006753 in OEIS. )

W. L. McDaniel proved in 1987 that exist infinitely many Smith numbers. While there are still among the first 1,000 numbers about 5 percent Smith numbers ( ie 49), there are about 3 percent of the first million and the first billion numbers a total of approximately 2.5 percent.

Special Smith numbers

Two successive Smith - numbers (eg 728 and 729, or 2964 and 2965 ) Smith Brothers are called. It is unknown how many of Smith Brothers exist. The smallest Smith Drilling form 73615, 73616, 73617, the smallest Vierling the numbers 4463535, 4463536, 4463537, 4463538th

Smith - numbers can be constructed from Repunits Rn. That's the largest known Smith - number:

In which

History

The Smith figures were named Albert Wilansky at Lehigh University. He noticed the special feature of the phone number of his brother Harold Smith. 4937775 (4937775 = 3 × 5 × 5 × 65837 → 4 9 3 7 7 7 5 = 3 5 5 6 5 8 3 7 = 42 )