Smith number


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.

Prime numbersMathworldPlanetmath are not considered, since it is obvious that all of them satisfy the condition given above.

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 (sequence A006753 in the OEIS).

There are infinitely many Smith numbers that are also palindromic numbersMathworldPlanetmath.

Smith numbers were named by Albert Wilansky of Lehigh University for his brother-in-law Harold Smith whose phone number (493-7775) was the first noticed Smith number.

Smith numbers can be constructed from factored repunitsMathworldPlanetmath. The largest known Smith number is (as of 2005) 9×R1031×(104594+3×102297+1)1476×103913210 where R1031=21032-1.

Title Smith number
Canonical name SmithNumber
Date of creation 2013-03-22 16:19:32
Last modified on 2013-03-22 16:19:32
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 4
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A63