A Smith number is a composite integer with the property that the sum of its digits is the same as the sum of the digits of its prime factors. For example, 16940 is Smith because 16940 = 2 x 2 x 5 x 7 x 11 x 11 and 1 + 6 + 9 + 4 + 0 = 2 + 2 + 5 + 7 + 1 + 1 + 1 + 1. The numbers are named after mathematician Albert Wilansky's brother-in-law, whose phone number is a Smith number.

An assortment of Smith facts:

  • The first ten Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, and 265.
  • There are an infinite number of Smith numbers.
  • Many products of repunit primes are Smith. In fact, if r is a repunit prime, then 1540r, 1720r, 25228r, and various other multiples are guaranteed to be Smith numbers.
  • Let p be prime, and define Dn(p) to be the number of times that the digit n appears in p. Then if D1(p) - D8(p) + 2 x (D2(p) - D7(p))  + 3 x (D3(p) - D6(p)) + 4 x (D4(p) - D5(p)) = 2, then 2p is Smith. For example, the Smith number 166 (= 2 x 83) is of this form: 83 is prime, and D3(83) = D8(83) = 1 whilst all other digits appear 0 times, reducing the formula above to 3 x D3(83) - D8(83) = 2.

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