Smith number

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.^[1] 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 OEIS)

Smith numbers were named by Albert Wilansky of Lehigh University.^[2] 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.^[2][3] 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 OEIS)

Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called Smith brothers.^[4] It is not known how many Smith brothers there are. The starting elements of the smallest Smith n-tuple for n=1,2,… are:^[5]

4, 728, 73615, 4463535, 15966114, 2050918644, 164736913905, … (sequence A059754 in 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 ×10^3913210

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

Notes

1. 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.
2. Sándor & Crstici (2004) p.383
3. McDaniel, Wayne (1987). "The existence of infinitely many k-Smith numbers". Fibonacci Quarterly 25 (1): 76–80. Zbl 0608.10012.
4. Sándor & Crstici (2004) p.384
5. Shyam Sunder Gupta. "Fascinating Smith Numbers".

References

• Gardner, Martin (1988). Penrose Tiles to Trapdoor Ciphers. pp. 299–300.
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.

External links

• Weisstein, Eric W., "Smith Number", MathWorld.
• Shyam Sunder Gupta, Fascinating Smith numbers.
• Copeland, Ed. "4937775 – Smith Numbers". Numberphile. Brady Haran.