 # Smith number

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A Smith number is a composite number for which, in a given base (in base 10 by default), the sum of its digits is equal to the sum of the digits in its prime factorization. For example, 378 = 2 × 3 × 3 × 3 × 7 is a Smith number since 3 + 7 + 8 = 2 + 3 + 3 + 3 + 7. In this definition the factors are treated as digits: for example, 22 factors to 2 × 11 and yields three digits: 2, 1, 1. Therefore 22 is a Smith number because 2 + 2 = 2 + 1 + 1.

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 the OEIS)

Smith numbers were named by Albert Wilansky of Lehigh University. 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 = 42.

## Properties

W.L. McDaniel in 1987 proved that there are infinitely many Smith numbers. The number of Smith numbers below 10n for n=1,2,… is:

1, 6, 49, 376, 3294, 29928, 278411, 2632758, 25154060, 241882509, … (sequence A104170 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 for n=1,2,… are:

4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, … (sequence A059754 in the OEIS)

Smith numbers can be constructed from factored repunits. The largest known Smith number as of 2010 is:

9 × R1031 × (104594 + 3×102297 + 1)1476 ×103913210

where R1031 is a repunit equal to (101031−1)/9.

## References

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