Practical number

In number theory, a practical number or panarithmic number[1] is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . 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.

Demonstration of the practicality of the number 12

The sequence of practical numbers (sequence A005153 in the 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 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." 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 numbersEdit

The original characterisation by Srinivasan (1948) stated that a practical number cannot be a deficient 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   is   with   and  , then Srinivasan's statement can be expressed by the inequality

 
. In other words, the ordered sequence of all divisors   of a practical number has to be a complete sub-sequence.

This partial characterization was extended and completed by Stewart (1954) and Sierpiński (1955) who showed that it is straightforward to determine whether a number is practical from its prime factorization. A positive integer greater than one with prime factorization   (with the primes in sorted order  ) is practical if and only if each of its prime factors   is small enough for   to have a representation as a sum of smaller divisors. For this to be true, the first prime   must equal 2 and, for every i from 2 to k, each successive prime   must obey the inequality

 

where   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.

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   as a sum of divisors of  , because if the inequality failed to be true then even adding together all the smaller divisors would give a sum too small to reach  . In the other direction, the condition is sufficient, as can be shown by induction. More strongly, if the factorization of   satisfies the condition above, then any   can be represented as a sum of divisors of  , by the following sequence of steps:[4]

  • By induction on  , it can be shown that  . Hence  .
  • Since the internals   cover   for  , there are such a   and some   such that  .
  • Since   and   can be shown by induction to be practical, we can find a representation of q as a sum of divisors of  .
  • Since  , and since   can be shown by induction to be practical, we can find a representation of r as a sum of divisors of  .
  • The divisors representing r, together with   times each of the divisors representing q, together form a representation of m as a sum of divisors of  .

PropertiesEdit

  • The only odd practical number is 1, because if   is an odd number greater than 2, then 2 cannot be expressed as the sum of distinct divisors of  . 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] More strongly the least common multiple of any two practical numbers is also a practical number. Equivalently, the set of all practical numbers is closed under multiplication.
  • From the above characterization by Stewart and Sierpiński it can be seen that if   is a practical number and   is one of its divisors then   must also be a practical number.
  • In the set of all practical numbers there is a primitive set of practical numbers. A primitive practical number is either practical and squarefree or practical and when divided by any of its prime factors whose factorization exponent is greater than 1 is no longer practical. The sequence of primitive practical numbers (sequence A267124 in the OEIS) begins
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 ...
  • Every positive integer has a practical multiple. For instance, for every integer  , its multiple   is practical.[6]

Relation to other classes of numbersEdit

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

  • From the above properties with   a practical number and   one of its divisors (that is,  ) then   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.
  • Every power 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 even perfect number is also a practical number.[7] This follows from Leonhard Euler's result that an even perfect number must have the form  . 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.
  • Every primorial (the product of the first   primes, for some  ) is practical.[7] For the first two primorials, two and six, this is clear. Each successive primorial is formed by multiplying a prime number   by a smaller primorial that is divisible by both two and the next smaller prime,  . By Bertrand'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   primes must also be practical. This includes Ramanujan's highly composite numbers (numbers with more divisors than any smaller positive integer) as well as the factorial numbers.[7]

Practical numbers and Egyptian fractionsEdit

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

 

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 rational number   has an Egyptian fraction representation with   terms. The proof involves finding a sequence of practical numbers   with the property that every number less than   may be written as a sum of   distinct divisors of  . Then,   is chosen so that  , and   is divided by   giving quotient   and remainder  . It follows from these choices that  . Expanding both numerators on the right hand side of this formula into sums of divisors of   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   has an Egyptian fraction representation in which the largest denominator is  .

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

Analogies with prime numbersEdit

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  .[9] Melfi also showed[10] that there are infinitely many practical Fibonacci numbers (sequence A124105 in the 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   for any positive real  , a result analogous to Legendre's conjecture for primes. Moreover, for all sufficiently large  , the interval   contains many practical numbers.[11]

Let   count how many practical numbers are at most  . Margenstern (1991) conjectured that   is asymptotic to   for some constant  , 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. Improving on an estimate of Tenenbaum (1986), Saias (1997) found that   has order of magnitude  . Weingartner (2015) proved Margenstern's conjecture. We have[12]

 
where  [13] Thus the practical numbers are about 33.6% more numerous than the prime numbers. The exact value of the constant factor   is given by[14]
 
where   is the Euler–Mascheroni constant and   runs over primes.

As with prime numbers in an arithmetic progression, given two natural numbers   and  , we have[15]

 
The constant factor   is positive if, and only if, there is more than one practical number congruent to  . If  , then  . 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%.

NotesEdit

ReferencesEdit

  • Eppstein, David (2021), "Egyptian fractions with denominators from sequences closed under doubling", Journal of Integer Sequences, 24: 21.8.8, arXiv:2109.12217
  • Erdős, Paul; Loxton, J. H. (1979), "Some problems in partitio numerorum", Journal of the Australian Mathematical Society, Series A, 27 (3): 319–331, doi:10.1017/S144678870001243X.
  • Heyworth, M. R. (1980), "More on panarithmic numbers", New Zealand Math. Mag., 17 (1): 24–28. As cited by Margenstern (1991).
  • Hausman, Miriam; Shapiro, Harold N. (1984), "On practical numbers", Communications on Pure and Applied Mathematics, 37 (5): 705–713, doi:10.1002/cpa.3160370507, MR 0752596.
  • Margenstern, Maurice (1984), "Résultats et conjectures sur les nombres pratiques", Comptes Rendus de l'Académie des Sciences, Série I, 299 (18): 895–898. As cited by Margenstern (1991).
  • Margenstern, Maurice (1991), "Les nombres pratiques: théorie, observations et conjectures", Journal of Number Theory, 37 (1): 1–36, doi:10.1016/S0022-314X(05)80022-8, MR 1089787.
  • Melfi, Giuseppe (1995), "A survey on practical numbers", Rend. Sem. Mat. Univ. Pol. Torino, 53 (4): 347–359.
  • Melfi, Giuseppe (1996), "On two conjectures about practical numbers", Journal of Number Theory, 56 (1): 205–210, doi:10.1006/jnth.1996.0012, MR 1370203.
  • Mitrinović, Dragoslav S.; Sándor, József; Crstici, Borislav (1996), "III.50 Practical numbers", Handbook of number theory, Volume 1, Mathematics and its Applications, vol. 351, Kluwer Academic Publishers, pp. 118–119, ISBN 978-0-7923-3823-9.
  • Pomerance, C.; Weingartner, A. (2021), "On primes and practical numbers", Ramanujan Journal, arXiv:2007.11062, doi:10.1007/s11139-020-00354-y.
  • Robinson, D. F. (1979), "Egyptian fractions via Greek number theory", New Zealand Math. Mag., 16 (2): 47–52. As cited by Margenstern (1991) and Mitrinović, Sándor & Crstici (1996).
  • Saias, Eric (1997), "Entiers à diviseurs denses, I", Journal of Number Theory, 62 (1): 163–191, doi:10.1006/jnth.1997.2057, MR 1430008.
  • Sigler, Laurence E. (trans.) (2002), Fibonacci's Liber Abaci, Springer-Verlag, pp. 119–121, ISBN 0-387-95419-8.
  • Sierpiński, Wacław (1955), "Sur une propriété des nombres naturels", Annali di Matematica Pura ed Applicata, 39 (1): 69–74, doi:10.1007/BF02410762.
  • Srinivasan, A. K. (1948), "Practical numbers" (PDF), Current Science, 17: 179–180, MR 0027799, archived from the original (PDF) on 2016-03-05.
  • Stewart, B. M. (1954), "Sums of distinct divisors", American Journal of Mathematics, The Johns Hopkins University Press, 76 (4): 779–785, doi:10.2307/2372651, JSTOR 2372651, MR 0064800.
  • Tenenbaum, G. (1986), "Sur un problème de crible et ses applications", Ann. Sci. École Norm. Sup. (4), 19 (1): 1–30, MR 0860809.
  • Tenenbaum, G.; Yokota, H. (1990), "Length and denominators of Egyptian fractions", Journal of Number Theory, 35 (2): 150–156, doi:10.1016/0022-314X(90)90109-5, MR 1057319.
  • Vose, M. (1985), "Egyptian fractions", Bulletin of the London Mathematical Society, 17 (1): 21, doi:10.1112/blms/17.1.21, MR 0766441.
  • Weingartner, A. (2015), "Practical numbers and the distribution of divisors", The Quarterly Journal of Mathematics, 66 (2): 743–758, arXiv:1405.2585, doi:10.1093/qmath/hav006.
  • Weingartner, A. (2019), "On the constant factor in several related asymptotic estimates", Mathematics of Computation, 88 (318): 1883–1902, arXiv:1705.06349, doi:10.1090/mcom/3402.
  • Weingartner, A. (2020), "The constant factor in the asymptotic for practical numbers", International Journal of Number Theory, 16 (3): 629–638, arXiv:1906.07819, doi:10.1142/S1793042120500311.
  • Weingartner, A. (2021), "An extension of the Siegel-Walfisz theorem", Proceedings of the American Mathematical Society, arXiv:2011.06627, doi:10.1090/proc/15607.
  • Weingartner, A. (2022), "Somewhat smooth numbers in short intervals", Ramanujan Journal, arXiv:2105.13568, doi:10.1007/s11139-022-00552-w.

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