Smith numbers

A Smith number has the same digital sum as its prime factors (with multiplicities).

<problem—different digital sums. There are at least three kinds: the simple sum of digits, used infra; the modulo-10 sum of the digits, described here; and the iterated sum (keep doing the first kind of digital sum until a fixed point is reached).>

• 4=2×2; digital sums, 4=2+2
• 666=2×3×3×37; digital sums, 6+6+6=2+3+3+3+7=18
• 1776=2×2×2×2×3×37; digital sums, 1+7+7+6=2+2+2=2+3+3+7=21

There are 29928 Smiths less than a million. It is believed that about 3% of any million consecutive integers are Smith numbers.

There are an infinite number of palindromatic Smiths.

Consecutive Smith numbers (eg 728, 729; 2964, 2965) are called Smith brothers. It is not known <update?> 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).

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