Demonstration of the practicality of the number 12
Demonstration of the practicality of the number 12

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.

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 numbers

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]

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 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 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 .[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%.

Notes

References

  • 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.