A Smith number is a number which in a given base, the sum of its digits is equal to the sum of the digits in its factorization. (In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed). For example, 202 is a Smith number, since 2 + 0 + 2 = 4, and its factorization is 2 × 101, and 2 + 1 + 0 + 1 = 4. In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ... In mathematics, a square-free integer is one divisible by no perfect square, except 1. ...

Prime numbers are not considered, since it is obvious that all of them satisfy the condition given above. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

In base 10, the first few Smith numbers are

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, 985, 1086 4 (four) is a number, numeral, and glyph. ... 22 (twenty-two) is the natural number following 21 and preceding 23. ... 27 (twenty-seven) is the natural number following 26 and preceding 28. ... 58 is the natural number following 57 and preceding 59. ... 85 is the natural number following 84 and preceding 86. ... 94 is the natural number following 93 and preceding 95. ... 121 is the natural number following 120 and preceding 122. ... 666 is the Number of the Beast in most versions of the Christian Bible, in the Book of Revelation. ...

W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers. There are 29,928 Smith numbers below one million. It is believed that about 3% of any million consecutive integers are Smith numbers.

There are an infinite number of palindromic Smith numbers. A palindrome is a word, phrase, number or any other sequence of units (like a strand of DNA) which has the property of reading the same in either direction (the adjustment of spaces between letters is generally permitted). ...

Consecutive Smith numbers (for example, 728 and 729, 2964 and 2965) are called Smith brothers. It is not known how many Smith brothers there are.

Smith numbers were named by Albert Wilansky of Lehigh University for his brother-in-law Harold Smith, who noticed the property in his phone number (4937775). Lehigh University is a university based in Bethlehem, Pennsylvania (within 75 miles (120 km) of Philadelphia and New York City). ...

External links

• Article on Smith Numbers at MathWorld

References

• Martin Gardner, Penrose Tiles to Trapdoor Ciphers, 1988, p299–300