Smith number

A Smith number is a composite number where the sum of its digits is equal to the sum of all the digits of its prime factors . The prime factors are given without exponents and are accordingly repeated in the product representation as often as necessary. ( 378 = 2 × 3 × 3 × 3 × 7 instead of 378 = 2 × 3 ³ × 7. )

example

The digit sum of the number 166 is 1 + 6 + 6 = 13.

166 = 2 × 83, the sum of the digits of their prime factors is therefore 2 + 8 + 3, which also results in 13.

So 166 is a Smith number.

Smith numbers in the decimal system

The first Smith numbers in the decimal system are 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378. (Follow A006753 in OEIS .)

WL McDaniel proved in 1987 that there are infinitely many Smith numbers. While the first 1,000 numbers contain around 5 percent Smith numbers (namely 49), the first million numbers around 3 percent and the first billion numbers a total of around 2.5 percent.

Special Smith numbers

Two consecutive Smith numbers (for example, 728 and 729, or 2964 and 2965) are called the Smith brothers. It is unknown how many Smith brothers exist. The smallest Smith triplet is formed by 73615, 73616, 73617, and the smallest quadruple is formed by the numbers 4463535, 4463536, 4463537, 4463538.

Smith numbers can be constructed from repunits R _n . This is the largest known Smith number:

${\ displaystyle 9 \ cdot R_ {1031} \ cdot (10 ^ {4594} +3 \ cdot 10 ^ {2397} +1) ^ {1476} \ cdot 10 ^ {3913210}}$

in which ${\ displaystyle R_ {1031} = {\ frac {10 ^ {1031} -1} {9}}}$

history

The Smith Numbers got their name from Albert Wilansky at Lehigh University . He noticed the special feature of the telephone number of his brother-in-law Harold Smith: 4937775. (4937775 = 3 × 5 × 5 × 65837 → 4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42.)

Individual evidence

^ Wayne McDaniel: The existence of infinitely many k-Smith numbers . In: Fibonacci Quarterly . Vol. 25, No. 1 , 1987, pp. 76-80 .
↑ OEIS: Number of Smith numbers below 10 ^ n.
↑ Shyam Sunder Gupta: Fascinating Smith Numbers . Retrieved February 22, 2014
^ Wolfram MathWorld: Smith Numbers . Retrieved February 22, 2014

literature

Martin Gardner : Penrose Tiles to Trapdoor Ciphers 1988, pp. 299-300.

Web links

Follow A006753 in OEIS
Eric W. Weisstein : Smith Number . In: MathWorld (English).
Shyam Sunder Gupta, Fascinating Smith Numbers .

[1] Wayne McDaniel: The existence of infinitely many k-Smith numbers . In: Fibonacci Quarterly . Vol. 25, No. 1 , 1987, pp. 76-80 .

[2] OEIS: Number of Smith numbers below 10 ^ n.

[3] Shyam Sunder Gupta: Fascinating Smith Numbers . Retrieved February 22, 2014

[4] Wolfram MathWorld: Smith Numbers . Retrieved February 22, 2014