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. It's important to remember that, by definition, the factors are treated as digits. For example, 22 in base 10 factors to 2 × 11 and yields three digits: 2, 1, 1. Therefore 22 is a Smith number because 2 + 2 = 2 + 1 + 1.

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.^[1] Of the first million positive integers, 29,928 (or about 2.99%) are Smith numbers, while about 2.41% of the first 10¹⁰ positive integers are Smith numbers.

There are infinitely many palindromic Smith numbers.^{[citation needed]}

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. He noticed the property in the phone number (4937775) of his brother-in-law Harold Smith. 4937775 = 3 × 5 × 5 × 65837, and 4+9+3+7+7+7+5 = 3 + 5 + 5 + 6+5+8+3+7 = 42.

Smith numbers can be constructed from factored repunits. The largest known Smith number is^{[update needed]}

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

where R₁₀₃₁ = (10¹⁰³¹−1)/9.