Smith number

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A Smith number is a composite number for which, in a given base, the sum of its digits is equal to the sum of the digits in its prime 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, 378 = 2 × 3 × 3 × 3 × 7 is a base 10 Smith number, since 3 + 7 + 8 = 2 + 3 + 3 + 3 + 7.

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 (sequence A006753 in OEIS)

W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers. There are 29,928 Smith numbers below one million. Practical investigation has shown the density of Smith numbers falling to 2.41% by 10¹⁰.

There are infinitely many palindromic Smith numbers.

Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called Smith brothers. It is not known how many Smith brothers there are. The smallest Smith triple is (73615, 73616, 73617), quads (4463535, 4463536, 4463537, 4463538), quints (15966114,...) and 6 consecutive Smith numbers (2050918644,...). [1]

Smith numbers were named by Albert Wilansky of Lehigh University for his brother-in-law Harold Smith whose phone number (4937775) was the first noticed Smith number.

Smith numbers can be constructed from factored repunits. The largest known Smith number is (as of 2005)

9 × R₁₀₃₁ × (10⁴⁵⁹⁴ + 3 × 10²²⁹⁷ + 1)¹⁴⁷⁶ × 10^3913210

where R₁₀₃₁ = (10¹⁰³¹−1)/9.