Type of composite integer
In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. 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.
Smith numbers were named by Albert Wilansky of Lehigh University, as he noticed the property in the phone number (493-7775) of his brother-in-law Harold Smith:
 4937775 = 3 · 5 · 5 · 65837
while 
 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + (6 + 5 + 8 + 3 + 7)
in base 10. 
Mathematical definition.
Let formula_1 be a natural number. For base formula_2, let the function formula_3 be the digit sum of formula_1 in base formula_5. A natural number formula_1 with prime factorization
formula_7
is a Smith number if
formula_8
Here the exponent formula_9 is the multiplicity of formula_10 as a prime factor of formula_1 (also known as the "p"-adic valuation of formula_1).
For example, in base 10, 378 = 21 · 33 · 71 is a Smith number since 3 + 7 + 8 = 2 · 1 + 3 · 3 + 7 · 1, and 22 = 21 · 111 is a Smith number, because 2 + 2 = 2 · 1 + (1 + 1) · 1.
The first few Smith numbers in base 10 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. (sequence in the OEIS)
Properties.
W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers.
The number of Smith numbers in base 10 below 10"n" for "n" = 1, 2, ... is given by
 1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, ... (sequence in the OEIS).
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 starting elements of the smallest Smith "n"-tuple (meaning "n" consecutive Smith numbers) in base 10 for "n" = 1, 2, ... are
 4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, ... (sequence in the OEIS).
Smith numbers can be constructed from factored repunits. As of 2010[ [update]], the largest known Smith number in base 10 is
9 × R1031 × (104594 + 3×102297 + 1)1476 ×103913210
where R1031 is the base 10 repunit (101031 − 1)/9.
Notes.
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