Innumber theory, apractical number orpanarithmic number^[1] is a positive integer${\displaystyle n}$ such that all smaller positive integers can be represented as sums of distinctdivisors of${\displaystyle n}$. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.

The sequence of practical numbers (sequenceA005153 in theOEIS) begins

Practical numbers were used byFibonacci in hisLiber Abaci (1202) in connection with the problem of representing rational numbers asEgyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.^[2]

The name "practical number" is due toSrinivasan (1948). He noted that "the subdivisions of money, weights, and measures involve numbers like 4, 12, 16, 20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10." His partial classification of these numbers was completed byStewart (1954) andSierpiński (1955). This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every evenperfect number and everypower of two is also a practical number.

Practical numbers have also been shown to be analogous withprime numbers in many of their properties.^[3]

The original characterisation bySrinivasan (1948) stated that a practical number cannot be adeficient number, that is one of which the sum of all divisors (including 1 and itself) is less than twice the number unless the deficiency is one. If the ordered set of all divisors of the practical number${\displaystyle n}$ is${\displaystyle d_1,d_2,...,d_j}$ with${\displaystyle d_1=1}$ and${\displaystyle d_j=n}$, then Srinivasan's statement can be expressed by the inequality$${\displaystyle 2n\le 1+\sum _{i=1}^j{d}_i.}$$In other words, the ordered sequence of all divisors of a practical number has to be acomplete sub-sequence.

This partial characterization was extended and completed byStewart (1954) andSierpiński (1955) who showed that it is straightforward to determine whether a number is practical from itsprime factorization.A positive integer greater than one with prime factorization${\displaystyle n=p_1^{{\alpha }_1}...p_k^{{\alpha }_k}}$ (with the primes in sorted order) is practical if and only if each of its prime factors${\displaystyle p_i}$ is small enough for${\displaystyle p_i-1}$ to have a representation as a sum of smaller divisors. For this to be true, the first prime${\displaystyle p_1}$ must equal 2 and, for everyi from 2 to k, each successive prime${\displaystyle p_i}$ must obey the inequality

${\displaystyle p_i\le 1+\sigma (p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\dots p_{i-1}^{{\alpha }_{i-1}})=1+\sigma (p_1^{{\alpha }_1})\sigma (p_2^{{\alpha }_2})\dots \sigma (p_{i-1}^{{\alpha }_{i-1}})=1+\prod _{j=1}^{i-1}\frac{p_j^{{\alpha }_j+1}-1}{p_j-1},}$

where${\displaystyle \sigma (x)}$ denotes thesum of the divisors ofx. For example, 2 × 3² × 29 × 823 = 429606 is practical, because the inequality above holds for each of its prime factors: 3 ≤ σ(2) + 1 = 4, 29 ≤ σ(2 × 3²) + 1 = 40, and 823 ≤ σ(2 × 3² × 29) + 1 = 1171.

The condition stated above is necessary and sufficient for a number to be practical. In one direction, this condition is necessary in order to be able to represent${\displaystyle p_i-1}$ as a sum of divisors of${\displaystyle n}$, because if the inequality failed to be true then even adding together all the smaller divisors would give a sum too small to reach${\displaystyle p_i-1}$. In the other direction, the condition is sufficient, as can be shown by induction. More strongly, if the factorization of${\displaystyle n}$ satisfies the condition above, then any${\displaystyle m\le \sigma (n)}$ can be represented as a sum of divisors of${\displaystyle n}$, by the following sequence of steps:^[4]

• By induction on${\displaystyle j\in [1,{\alpha }_k]}$, it can be shown that${\displaystyle p_k^j\le 1+\sigma (n/p_k^{{\alpha }_k-(j-1)})}$. Hence${\displaystyle p_k^{{\alpha }_k}\le 1+\sigma (n/p_k)}$.
• Since the internals${\displaystyle [q{p}_k^{{\alpha }_k},q{p}_k^{{\alpha }_k}+\sigma (n/p_k)]}$ cover${\displaystyle [1,\sigma (n)]}$ for${\displaystyle 1\le q\le \sigma (n/p_k^{{\alpha }_k})}$, there are such a${\displaystyle q}$ and some${\displaystyle r\in [0,\sigma (n/p_k)]}$ such that${\displaystyle m=q{p}_k^{{\alpha }_k}+r}$.
• Since${\displaystyle q\le \sigma (n/p_k^{{\alpha }_k})}$ and${\displaystyle n/p_k^{{\alpha }_k}}$ can be shown by induction to be practical, we can find a representation ofq as a sum of divisors of${\displaystyle n/p_k^{{\alpha }_k}}$.
• Since${\displaystyle r\le \sigma (n/p_k)}$, and since${\displaystyle n/p_k}$ can be shown by induction to be practical, we can find a representation ofr as a sum of divisors of${\displaystyle n/p_k}$.
• The divisors representingr, together with${\displaystyle p_k^{{\alpha }_k}}$ times each of the divisors representingq, together form a representation ofm as a sum of divisors of${\displaystyle n}$.

• The only odd practical number is 1, because if${\displaystyle n}$ is an odd number greater than 2, then 2 cannot be expressed as the sum of distinct divisorsof${\displaystyle n}$. More strongly,Srinivasan (1948) observes that other than 1 and 2, every practical number is divisible by 4 or 6 (or both).
• The product of two practical numbers is also a practical number.^[5] Equivalently, the set of all practical numbers is closed under multiplication. More strongly, theleast common multiple of any two practical numbers is also a practical number.
• From the above characterization by Stewart and Sierpiński it can be seen that if${\displaystyle n}$ is a practical number and${\displaystyle d}$ is one of its divisors then${\displaystyle n\cdot d}$ must also be a practical number. Furthermore, a practical number multiplied by power combinations of any of its divisors is also practical.
• In the set of all practical numbers there is a primitive set of practical numbers. A primitive practical number is either practical andsquarefree or practical and when divided by any of its prime factors whosefactorization exponent is greater than 1 is no longer practical. The sequence of primitive practical numbers (sequenceA267124 in theOEIS) begins

• Every positive integer has a practical multiple. For instance, for every integer${\displaystyle n}$, its multiple${\displaystyle 2^{\lfloor {\log }_{2}n\rfloor }n}$ is practical.^[6]
• Every odd prime has a primitive practical multiple. For instance, for every odd prime${\displaystyle p}$, its multiple${\displaystyle 2^{\lfloor {\log }_{2}p\rfloor }p}$ is primitive practical. This is because${\displaystyle 2^{\lfloor {\log }_{2}p\rfloor }p}$ is practical^[6] but when divided by 2 is no longer practical. A good example is aMersenne prime of the form${\displaystyle 2^p-1}$. Its primitive practical multiple is${\displaystyle 2^{p-1}(2^p-1)}$ which is an evenperfect number.

Several other notable sets of integers consist only of practical numbers:

• From the above properties with${\displaystyle n}$ a practical number and${\displaystyle d}$ one of its divisors (that is,${\displaystyle d|n}$) then${\displaystyle n\cdot d}$ must also be a practical number therefore six times every power of 3 must be a practical number as well as six times every power of 2.
• Everypower of two is a practical number.^[7] Powers of two trivially satisfy the characterization of practical numbers in terms of their prime factorizations: the only prime in their factorizations,p₁, equals two as required.
• Every evenperfect number is also a practical number.^[7] This follows fromLeonhard Euler's result that an even perfect number must have the form${\displaystyle 2^{k-1}(2^k-1)}$. The odd part of this factorization equals the sum of the divisors of the even part, so every odd prime factor of such a number must be at most the sum of the divisors of the even part of the number. Therefore, this number must satisfy the characterization of practical numbers. A similar argument can be used to show that an even perfect number when divided by 2 is no longer practical. Therefore, every even perfect number is also a primitive practical number.
• Everyprimorial (the product of the first${\displaystyle i}$ primes, for some${\displaystyle i}$) is practical.^[7] For the first two primorials, two and six, this is clear. Each successive primorial is formed by multiplying a prime number${\displaystyle p_i}$ by a smaller primorial that is divisible by both two and the next smaller prime,${\displaystyle p_{i-1}}$. ByBertrand's postulate,, so each successive prime factor in the primorial is less than one of the divisors of the previous primorial. By induction, it follows that every primorial satisfies the characterization of practical numbers. Because a primorial is, by definition, squarefree it is also a primitive practical number.
• Generalizing the primorials, any number that is the product of nonzero powers of the first${\displaystyle k}$ primes must also be practical. This includesRamanujan'shighly composite numbers (numbers with more divisors than any smaller positive integer) as well as thefactorial numbers.^[7]

If${\displaystyle n}$ is practical, then anyrational number of the form${\displaystyle m/n}$ with may be represented as a sum$\sum d_i/n$ where each${\displaystyle d_i}$ is a distinct divisor of${\displaystyle n}$. Each term in this sum simplifies to aunit fraction, so such a sum provides a representation of${\displaystyle m/n}$ as anEgyptian fraction. For instance,$${\displaystyle \frac{13}{20}=\frac{10}{20}+\frac{2}{20}+\frac{1}{20}=\frac{1}{2}+\frac{1}{10}+\frac{1}{20}.}$$

Fibonacci, in his 1202 bookLiber Abaci^[2] lists several methods for finding Egyptian fraction representations of a rational number. Of these, the first is to test whether the number is itself already a unit fraction, but the second is to search for a representation of the numerator as a sum of divisors of the denominator, as described above. This method is only guaranteed to succeed for denominators that are practical. Fibonacci provides tables of these representations for fractions having as denominators the practical numbers 6, 8, 12, 20, 24, 60, and 100.

Vose (1985) showed that every rational number${\displaystyle x/y}$ has an Egyptian fraction representation with${\displaystyle O(\sqrt{\log y})}$ terms. The proof involves finding a sequence of practical numbers${\displaystyle n_i}$ with the property that every number less than${\displaystyle n_i}$ may be written as a sum of${\displaystyle O(\sqrt{\log n_{i-1}})}$ distinct divisors of${\displaystyle n_i}$. Then,${\displaystyle i}$ is chosen so that, and${\displaystyle x{n}_i}$ is divided by${\displaystyle y}$ giving quotient${\displaystyle q}$ and remainder${\displaystyle r}$. It follows from these choices that${\displaystyle \frac{x}{y}=\frac{q}{n_i}+\frac{r}{y{n}_i}}$. Expanding both numerators on the right hand side of this formula into sums of divisors of${\displaystyle n_i}$ results in the desired Egyptian fraction representation.Tenenbaum & Yokota (1990) use a similar technique involving a different sequence of practical numbers to show that every rational number${\displaystyle x/y}$ has an Egyptian fraction representation in which the largest denominator is${\displaystyle O(y{\log }^{2}y/\log \log y)}$.

According to a September 2015 conjecture byZhi-Wei Sun,^[8] every positive rational number has an Egyptian fraction representation in which every denominator is a practical number. The conjecture was proved byDavid Eppstein (2021).

One reason for interest in practical numbers is that many of their properties are similar to properties of theprime numbers. Indeed, theorems analogous toGoldbach's conjecture and thetwin prime conjecture are known for practical numbers: every positive even integer is the sum of two practical numbers, and there exist infinitely many triples of practical numbers${\displaystyle (x-2,x,x+2)}$.^[9]Melfi also showed^[10] that there are infinitely many practicalFibonacci numbers (sequenceA124105 in theOEIS); and Sanna^[11] proved that at least${\displaystyle Cn/\log n}$ of the first${\displaystyle n}$ terms of everyLucas sequence are practical numbers, where${\displaystyle C>0}$ is a constant and${\displaystyle n}$ is sufficiently large. The analogous questions of the existence of infinitely manyFibonacci primes, or prime in a Lucas sequence, are open.Hausman & Shapiro (1984) showed that there always exists a practical number in the interval${\displaystyle [x^2,(x+1{)}^2)]}$ for any positive real${\displaystyle x}$, a result analogous toLegendre's conjecture for primes. Moreover, for all sufficiently large${\displaystyle x}$, the interval${\displaystyle [x-x^{0.4872},x]}$ contains many practical numbers.^[12]

Let${\displaystyle p(x)}$ count how many practical numbers are atmost${\displaystyle x}$.Margenstern (1991) conjectured that${\displaystyle p(x)}$ is asymptotic to${\displaystyle cx/\log x}$ for some constant${\displaystyle c}$, a formula which resembles theprime number theorem, strengthening the earlier claim ofErdős & Loxton (1979) that the practical numbers have density zero in the integers.Improving on an estimate ofTenenbaum (1986),Saias (1997) found that${\displaystyle p(x)}$ has order of magnitude${\displaystyle x/\log x}$.Weingartner (2015) proved Margenstern's conjecture. We have^[13]$${\displaystyle p(x)=\frac{cx}{\log x}\left(1+O\left(\frac{1}{\log x}\right)\right),}$$where${\displaystyle c=\mathrm{1.33607...}}$^[14] Thus the practical numbers are about 33.6% more numerous than the prime numbers. The exact value of the constant factor${\displaystyle c}$ is given by^[15]$${\displaystyle c=\frac{1}{1-e^{-\gamma }}\sum _{n\text{practical}}\frac{1}{n}(\sum _{p\le \sigma (n)+1}\frac{\log p}{p-1}-\log n)\prod _{p\le \sigma (n)+1}\left(1-\frac{1}{p}\right),}$$where${\displaystyle \gamma }$ is theEuler–Mascheroni constant and${\displaystyle p}$ runs over primes.

As with prime numbers in an arithmetic progression, given two natural numbers${\displaystyle a}$ and${\displaystyle q}$, we have^[16]$${\displaystyle |\{n\le x:n\text{ practical and }n\equiv amodq\}|=\frac{c_{q,a}x}{\log x}+O_q\left(\frac{x}{(\log x{)}^2}\right).}$$The constant factor${\displaystyle c_{q,a}}$ is positive if, and only if, there is more than one practical number congruent to${\displaystyle amodq}$.If${\displaystyle gcd(q,a)=gcd(q,b)}$, then${\displaystyle c_{q,a}=c_{q,b}}$. For example, about 38.26% of practical numbers have a last decimal digit of 0, while the last digits of 2, 4, 6, 8 each occur with the same relative frequency of 15.43%.

TheErdős–Kac theorem implies that for a large random integer${\displaystyle n}$, the number of prime factors of${\displaystyle n}$ (counted with or without multiplicity) follows an approximatenormal distribution with mean${\displaystyle \log \log n}$ and variance${\displaystyle \log \log n}$. The corresponding result for practical numbers^[17] implies that for a large random practical number${\displaystyle n}$, the number of prime factors is approximately normal with mean${\displaystyle C\log \log n}$ and variance${\displaystyle V\log \log n}$, where${\displaystyle C=1/(1-e^{-\gamma })=2.280\dots }$ and${\displaystyle V=0.414\dots }$. That is, most large integers${\displaystyle n}$ have about${\displaystyle \log \log n}$ prime factors, while most large practical numbers${\displaystyle n}$ have about${\displaystyle C\log \log n\approx 2.28\log \log n}$ prime factors.

As a consequence, most large integers${\displaystyle n}$ have${\displaystyle 2^{(1+o(1))\log \log n}=(\log n{)}^{0.693\dots }}$ divisors, while most large practical numbers${\displaystyle n}$ have${\displaystyle 2^{(C+o(1))\log \log n}=(\log n{)}^{1.580\dots }}$ divisors. In both cases, the average number of divisors is much larger than the typical number of divisors: for integers${\displaystyle n\le x}$, the average number of divisors is about${\displaystyle \log x}$, while for practical numbers${\displaystyle n\le x}$, it is about${\displaystyle (\log x{)}^{1.713\dots }}$.^[18]

The average value of the sum-of-divisors function${\displaystyle \sigma (n)}$, for integers${\displaystyle n\le x}$, as well as for practical numbers${\displaystyle n\le x}$, has order of magnitude${\displaystyle x}$.^[19]

• Tables of practical numbersArchived 2017-12-26 at theWayback Machine compiled by Giuseppe Melfi.
• Practical Number atPlanetMath.
• Weisstein, Eric W.,"Practical Number",MathWorld

• Integer sequences
• Egyptian fractions

• Articles with short description
• Short description is different from Wikidata
• Webarchive template wayback links