Practical number

Demonstration, with Cuisenaire rods, of the practicality of the number 12

In number theory, a practical number or panarithmic number[1] is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of 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 (sequence A005153 in OEIS) begins

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....

Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian 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 to Srinivasan (1948). He noted that the subdivision of money, weights and measures involved numbers like 4, 12, 16, 20 and 28 which are usually supposed to be so inconvenient as to deserve replacement by powers of 10. He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of these numbers that was completed by Stewart (1954) and Sierpiński (1955). This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every even perfect number and every power of two is also a practical number.

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

Characterization of practical numbers

As Stewart (1954) and Sierpiński (1955) showed, it is straightforward to determine whether a number is practical from its prime factorization. A positive integer greater than one with prime factorization n=p_1^{\alpha_1}...p_k^{\alpha_k} (with the primes in sorted order p_1<p_2<\dots<p_k) is practical if and only if each of its prime factors p_i is small enough for p_i-1 to have a representation as a sum of smaller divisors. For this to be true, the first prime p_1 must equal 2 and, for every i from 2 to k, each successive prime p_i must obey the inequality

p_i\leq1+\sigma(p_1^{\alpha_1}p_2^{\alpha_2}\dots p_{i-1}^{\alpha_{i-1}})=1+\prod_{j=1}^{i-1}\frac{p_j^{\alpha_j+1}-1}{p_j-1},

where \sigma(x) denotes the sum of the divisors of x. 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. This characterization extends a partial classification of the practical numbers given by Srinivasan (1948).

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 p_i-1 as a sum of divisors of 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 p_i-1. In the other direction, the condition is sufficient, as can be shown by induction.

More strongly, one can show that, if the factorization of n satisfies the condition above, then any m \le \sigma(n) can be represented as a sum of divisors of n, by the following sequence of steps:

Properties

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 ...

Relation to other classes of numbers

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

Practical numbers and Egyptian fractions

If n is practical, then any rational number of the form m/n may be represented as a sum ∑di/n where each di is a distinct divisor of n. Each term in this sum simplifies to a unit fraction, so such a sum provides a representation of m/n as an Egyptian fraction. For instance,

\frac{13}{20}=\frac{10}{20}+\frac{2}{20}+\frac{1}{20}=\frac12+\frac1{10}+\frac1{20}.

Fibonacci, in his 1202 book Liber 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 number x/y has an Egyptian fraction representation with \scriptstyle O(\sqrt{\log y}) terms. The proof involves finding a sequence of practical numbers ni with the property that every number less than ni may be written as a sum of \scriptstyle O(\sqrt{\log n_{i-1}}) distinct divisors of ni. Then, i is chosen so that ni  1 < y  ni, and xni is divided by y giving quotient q and remainder r. It follows from these choices that \scriptstyle\frac{x}{y}=\frac{q}{n_i}+\frac{r}{yn_i}. Expanding both numerators on the right hand side of this formula into sums of divisors of ni 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 number x/y has an Egyptian fraction representation in which the largest denominator is \scriptstyle O(\frac{y\log^2 y}{\log\log y}).

Analogies with prime numbers

One reason for interest in practical numbers is that many of their properties are similar to properties of the prime numbers. Indeed, theorems analogous to Goldbach's conjecture and the twin 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 x  2, x, x + 2.[6] Melfi also showed that there are infinitely many practical Fibonacci numbers (sequence A124105 in OEIS); the analogous question of the existence of infinitely many Fibonacci primes is open. Hausman & Shapiro (1984) showed that there always exists a practical number in the interval [x2,(x + 1)2] for any positive real x, a result analogous to Legendre's conjecture for primes.

Let be p(x) count how many practical numbers are at most x. Margenstern (1991) conjectured that p(x) is asymptotic to cx/log x for some constant c, a formula which resembles the prime number theorem, strengthening the earlier claim of Erdős & Loxton (1979) that the practical numbers have density zero in the integers. Saias (1997) proved that for suitable constants c1 and c2:

c_1\frac x{\log x}<p(x)<c_2\frac x{\log x},

Finally Weingartner (2015) proved Margenstern's conjecture showing that

p(x) = \frac{c x}{\log x}\left(1 + O\!\left(\frac{\log \log x}{\log x}\right)\right),

for x \geq 3 and some constant c > 0.

Notes

References

External links


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