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Smith Number

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Binary plot of Smith numbers

A Smith number is a composite number whose base-10 digit sum equals the sum of the base-10 digit sums of its prime factors, counted with multiplicity. (The primes are excluded since they trivially satisfy this condition). One example of a Smith number is the beast number

666=2·3·3·37,
(1)

since

6+6+6=2+3+3+(3+7)=18.
(2)

Another Smith number is

4937775=3·5·5·65837,
(3)

since

4+9+3+7+7+7+5=3+5+5+(6+5+8+3+7)=42.
(4)

The first few Smith numbers are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, ... (OEIS A006753). The corresponding digit sums are 4, 4, 9, 13, 13, 13, 4, 13, 4, 13, 13, 13, 13, ... (OEIS A050218). McDaniel (1987a) showed that there are an infinite number of Smith numbers.

A generalized k-Smith number is a composite number m satisfying S_p(m)=kS(m) for positive k, where S_p(m) is the sum of the base-10 digits of the prime factors with multiplicity and S(m) is the base-10 digit sum. The following table gives the first few k-Smith numbers for small integers and their inverses.

kOEISk-Smith numbers
1/3A0502256969, 19998, 36399, 39693, 66099, 69663, ...
1/2A05022488, 169, 286, 484, 598, 682, 808, 844, 897, ...
1A0067534, 22, 27, 58, 85, 94, 121, 166, 202, 265, ...
2A10439032, 42, 60, 70, 104, 152, 231, 315, 316, 322, ...
3A104391402, 510, 700, 1113, 1131, 1311, 2006, 2022, ...

A Smith number can be constructed from every factored repunit R_n (Hoffman 1998, pp. 205-206). Costello (2002, p. 370) records the earlier Smith-number record

9×R_(1031)(10^(4594)+3×10^(2297)+1)^(1476)×10^(3913210).
(5)

See also

Hoax Number, Monica Set, Perfect Number, Repunit, Smith Brothers, Suzanne Set

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References

Costello, P. "A New Largest Smith Number." Fib. Quart. 40, 369-371, 2002. https://doi.org/10.1080/00150517.2002.12428641.Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 99-100, 1989.Guy, R. K. "Smith Numbers." §B49 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 103-104, 1994.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, pp. 205-206, 1998.McDaniel, W. L. "The Existence of Infinitely Many k-Smith Numbers." Fib. Quart. 25, 76-80, 1987a.McDaniel, W. L. "Powerful k-Smith Numbers." Fib. Quart. 25, 225-228, 1987b.Oltikar, S. and Wayland, K. "Construction of Smith Numbers." Math. Mag. 56, 36-37, 1983.Pickover, C. A. "A Brief History of Smith Numbers." Ch. 104 in Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England: Oxford University Press, pp. 247-248, 2001.Sloane, N. J. A. Sequences A006753/M3582, A050218, A050224, A050225, A104390, and A104391 in "The On-Line Encyclopedia of Integer Sequences."Wilansky, A. "Smith Numbers." Two-Year College Math. J. 13, 21, 1982.Yates, S. "Special Sets of Smith Numbers." Math. Mag. 59, 293-296, 1986.Yates, S. "Smith Numbers Congruent to 4 (mod 9)." J. Recr. Math. 19, 139-141, 1987.

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Smith Number

Cite this as:

Weisstein, Eric W. "Smith Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SmithNumber.html

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