Practical number

In number theory , a practical number (from English practical number , also panarithmic number ) is a natural number with the property that every smaller number can be written as the sum of different real divisors of . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$

The mathematician AK Srinivasan first mentioned these numbers in 1948.

Examples

• The number has the proper factors and . All smaller numbers can be written as the sum of these real factors: ${\displaystyle n=28}$ ${\displaystyle 1,2,4,7}$ ${\displaystyle 14}$

${\displaystyle {\begin{array}{lllllll}\mathbf {1} =1,&\mathbf {2} =2,&\mathbf {3} =2+1,&\mathbf {4} =4,&\mathbf {5} =4+1,&\mathbf {6} =4+2,&\mathbf {7} =7,\\\mathbf {8} =7+1,&\mathbf {9} =7+2,&\mathbf {10} =7+2+1,&\mathbf {11} =7+4,&\mathbf {12} =7+4+1,&\mathbf {13} =7+4+2,&\mathbf {14} =14,\\\mathbf {15} =14+1,&\mathbf {16} =14+2,&\mathbf {17} =14+2+1,&\mathbf {18} =14+4,&\mathbf {19} =14+4+1,&\mathbf {20} =14+4+2,&\mathbf {21} =14+7,\\\mathbf {22} =14+7+1,&\mathbf {23} =14+7+2,&\mathbf {24} =14+7+2+1,&\mathbf {25} =14+7+4,&\mathbf {26} =14+7+4+1,&\mathbf {27} =14+7+4+2\\\end{array}}}$

Apparently one can write all numbers as the sum of these real divisors. So is a practical number. ${\displaystyle m<28}$ ${\displaystyle n=28}$

• The number has the proper factors and . One can write the following smaller numbers as the sum of these real factors: ${\displaystyle n=26}$ ${\displaystyle 1,2}$ ${\displaystyle 13}$

1 = 1, 2 = 2, 3 = 2 + 1, ...

But even the number 4 can no longer be written as the sum of these real divisors, because the requirement sum of real divisors that differ in pairs would have to be broken, namely 4 = 2 + 2. So the number is not a practical number. ${\displaystyle n=26}$

• The following numbers are the smallest practical numbers (which is 198th practical number ): ${\displaystyle n=1000}$

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252,…, 984, 990, 992, 1000, 1008,… ( continuation A005153 in OEIS )

properties

• A practical number is never a deficient number (so the sum of all real divisors of a practical number is never less than the number itself).
• All numbers in the form with are practical numbers. ${\displaystyle n=2^{k-1}(2^{k}-1)}$ ${\displaystyle k\in \mathbb {N} ,k\geq 2}$
• All even perfect numbers are practical numbers.
• Let be the product of powers (non-zero) of the first prime numbers. Then: ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k}$

${\displaystyle n\in \mathbb {N} }$ is a practical number.

In particular, every prime faculty (i.e. every product of the first prime numbers with ) is a practical number. ${\displaystyle k}$ ${\displaystyle k\in \mathbb {N} }$

• All composite numbers are practical numbers.
• All numbers of the form with , i.e. every power of two , are practical numbers. ${\displaystyle 2^{k}}$ ${\displaystyle k\in \mathbb {N} }$
• The only odd practical number is . ${\displaystyle n=1}$

Proof:

Suppose it is an odd number. Then that number has the proper factors and some major factors . However, this means that the number 2 cannot be represented as the sum of its real factors. Thus, it can not be a practical number. ${\displaystyle m>2}$ ${\displaystyle 1}$ ${\displaystyle k\geq 3}$ ${\displaystyle m}$ ${\displaystyle \Box }$

• Be a practical number. Then: ${\displaystyle n\geq 4}$

${\displaystyle n}$ is a multiple of 4 or 6 (or both numbers).

This statement has been proven by AK Srinivasan.

• The product of two practical numbers is again a practical number. The set of practical numbers is therefore closed with regard to multiplication.

Proof: see

• Be a practical number and a divisor of . Then: ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle d}$ ${\displaystyle n}$

${\displaystyle n\cdot d}$ is also a practical number.

• The least common multiple ( LCM for short ) of two practical numbers is again a practical number.
• There are infinitely many Fibonacci numbers , which are practical numbers. They are called practical Fibonacci numbers . The smallest are:

1, 2, 8, 144, 46368, 832040, 14930352, 267914296, 4807526976, 1548008755920, 498,454,011,879,264, 160500643816367088, 2880067194370816120, 51680708854858323072, 16641027750620563662096, 5358359254990966640871840 ... (sequence A124105 in OEIS )

• For every positive rational number : ${\displaystyle m\in \mathbb {Q} ^{+}}$

There is a representation of as the sum of finitely many stem fractions with pairwise different denominators , where each denominator is a practical number (a so-called division into Egyptian fractions ). ${\displaystyle m}$

In other words: be . Then: ${\displaystyle m\in \mathbb {Q} ^{+}}$

${\displaystyle m=\sum _{i=1}^{k}{\frac {1}{q_{i}}}}$with pairs of different practical numbers and ${\displaystyle q_{i}}$ ${\displaystyle k\in \mathbb {N} }$

Example 1:

Be . Then you can use a special procedure to break this number down into the four original fractions ${\displaystyle m={\frac {25}{31}}}$

${\displaystyle m={\frac {25}{31}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{18}}+{\frac {1}{1116}}}$

This division is a division into Egyptian fractions. You get the denominators and . In fact, these four denominators are all practical numbers. ${\displaystyle 2,4,18}$ ${\displaystyle 1116}$

Practical numbers are not always obtained immediately with this procedure, but there is always such a representation:

Example 2:

Be . Then you can use the same procedure to break this number down into the four original fractions ${\displaystyle m={\frac {14}{17}}}$

${\displaystyle m={\frac {14}{17}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {5}{68}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{14}}+{\frac {1}{476}}}$

This division is not a division into Egyptian fractions. You get the denominators and . While three of these four denominators are practical numbers ( and ), it is not a practical number. So you have to force the fraction to the next best denominator, which is a practical number, namely . Ultimately, you get the following Egyptian fractions: ${\displaystyle 2,4,14}$ ${\displaystyle 476}$ ${\displaystyle 2,4}$ ${\displaystyle 476}$ ${\displaystyle 14}$ ${\displaystyle {\frac {5}{68}}}$ ${\displaystyle {\frac {1}{16}}}$

${\displaystyle m={\frac {14}{17}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{16}}+{\frac {1}{96}}+{\frac {1}{1632}}}$

You get the denominators and . In fact, these five denominators are all practical numbers. ${\displaystyle 2,4,16,96}$ ${\displaystyle 1632}$

• Any even natural number can be represented as the sum of two practical numbers and . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$
• There are infinitely many triples (= three-tuples) of practical numbers. ${\displaystyle (n-2,n,n+2)}$
• There is always at least one practical number in the interval for all positive real numbers . ${\displaystyle [x^{2},(x+1)^{2}]}$ ${\displaystyle x\in \mathbb {R} ^{+}}$
• Let be the number of practical numbers that are less than or equal to. Then: ${\displaystyle p(x)}$ ${\displaystyle x\in \mathbb {N} }$

${\displaystyle p(x)={\frac {c\cdot x}{\log x}}\left(1+O\!\left({\frac {\log \log x}{\log x}}\right)\right)}$with and a constant ${\displaystyle x\geq 3}$ ${\displaystyle c>0}$

Unsolved problems

• It is believed that there are infinitely many five-tuples of practical numbers. ${\displaystyle (n-6,n-2,n,n+2,n+6)}$

Primitive practical numbers

A primitive practical number is a practical number that has one of the following two properties: ${\displaystyle n\in \mathbb {N} }$

• It is square-free .
• If you divide it by one of its prime factors whose exponent is greater than , the result can no longer be a practical number. ${\displaystyle 1}$

Examples

• The number , as shown above , is a practical number. It is not square-free because it is. If you divide it by one of its prime factors whose exponent is greater than , that is, by , you get which is not a practical number. Thus is a primitive practical number. ${\displaystyle n=28}$ ${\displaystyle n=28=2^{2}\cdot 7}$ ${\displaystyle 1}$ ${\displaystyle p=2}$ ${\displaystyle {\frac {28}{2}}=14}$ ${\displaystyle n=28}$
• The number is a practical number. Your prime factorization is . If you divide it by one of its prime factors whose exponent is greater than , that is, by or by , you get or , respectively , which are both still practical numbers. Thus it is not a primitive practical number. ${\displaystyle n=108}$ ${\displaystyle n=108=2^{2}\cdot 3^{3}}$ ${\displaystyle 1}$ ${\displaystyle p=2}$ ${\displaystyle p=3}$ ${\displaystyle {\frac {108}{2}}=54}$ ${\displaystyle {\frac {108}{3}}=36}$ ${\displaystyle n=108}$
• The smallest primitive practical numbers are the following (there are exactly 61 of them, which are less than or equal to 1000):

1, 2, 6, 20, 28, 30, 42, 66, 78, 88, 104, 140, 204, 210, 220, 228, 260, 272, 276, 304, 306, 308, 330, 340, 342, 348, 364, 368, 380, 390, 414, 460, 462, 464, 476, 496, 510, 522, 532, 546, 558, 570, 580, 620, 644, 666, 690, 714, 740, 744, 798, 812, 820, 858, 860, 868, 870, 888, 930, 966, 984, 1032, ... ( continuation A267124 in OEIS )

properties

• Every prime faculty (i.e. every product of the first prime numbers with ) is a primitive practical number. ${\displaystyle k}$ ${\displaystyle k\in \mathbb {N} }$

Web links

• Eric W. Weisstein : Practical Number . In: MathWorld (English).
• Melfi, Practical numbers

Individual evidence

1. ↑ ^{a b c d e f g} A. K. Srinivasan: Practical numbers. Current Science 17 , 1948, pp. 179-180 , accessed January 1, 2019 . 
2. ^ ^{A b} Ross Honsberger : Mathematical gems. Vieweg , 1981, p. 112 , accessed January 1, 2019 . 
3. ↑ Eric W. Weisstein : Practical Number . In: MathWorld (English).
4. ↑ Maurice Margenstern : Les nombres pratiques: théorie, observations et conjectures. (PDF) Journal of Number Theory 37 , 1991, pp. 1–36 , accessed on January 1, 2019 . 
5. ^ Zhi-Wei Sun : A Conjecture on Unit Fractions Involving Primes. (PDF) Nanjing University , Department of Mathematics, 2015, pp. 1–15 , accessed January 1, 2019 . 
6. ^ David Eppstein : Egyptian fractions with practical denominators. 2016, accessed January 1, 2019 . 
7. ^ ^{A b} Giuseppe Melfi : On two conjecture about practical numbers. (PDF) Journal of Number Theory 56 (1), 1996, pp. 205–210 , accessed on January 1, 2019 . 
8. ^ ^{A b c} Giuseppe Melfi : A survey on practical numbers. (PDF) rend. Sem. Mat. Univ. Pole. Torino , 53 , 1995, pp. 347-359 , accessed January 1, 2019 . 
9. ↑ Miriam Hausman , Harold N. Shapiro : On Practical Numbers. Communications on Pure and Applied Mathematics , 37 (5), 1984, pp. 705-713 , accessed January 1, 2019 . 
10. ^ Andreas Weingartner : Practical numbers and the distribution of divisors. The Quarterly Journal of Mathematics , 66 (2), 2015, pp. 743-758 , accessed January 1, 2019 .