Editing Practical number (section)

==References==
{{refbegin|30em}}
*{{citation|title=Egyptian fractions with denominators from sequences closed under doubling|first=David|last=Eppstein|journal=Journal of Integer Sequences|volume=24|page=21.8.8|url=https://cs.uwaterloo.ca/journals/JIS/VOL24/Eppstein/eppstein2.html|arxiv=2109.12217|year=2021}}
*{{citation
 | last1 = Erdős | first1 = Paul | author1-link = Paul Erdős
 | last2 = Loxton | first2 = J. H.
 | doi = 10.1017/S144678870001243X
 | journal = Journal of the Australian Mathematical Society, Series A
 | pages = 319–331
 | issue = 3
 | title = Some problems in partitio numerorum
 | volume = 27
 | year = 1979| doi-access = free
 }}.
*{{citation
 | last = Heyworth | first = M. R.
 | issue = 1
 | journal = New Zealand Math. Mag.
 | pages = 24–28
 | title = More on panarithmic numbers
 | volume = 17
 | year = 1980}}. As cited by {{harvtxt|Margenstern|1991}}.
*{{citation
 | last1 = Hausman | first1 = Miriam | last2 = Shapiro | first2 = Harold N.
 | title = On practical numbers
 | journal = [[Communications on Pure and Applied Mathematics]]
 | volume = 37
 | year = 1984
 | issue = 5
 | pages = 705–713
 | mr = 0752596
 | doi = 10.1002/cpa.3160370507}}.
*{{citation
 | last = Margenstern | first = Maurice
 | issue = 18
 | journal = Comptes Rendus de l'Académie des Sciences, Série I
 | pages = 895–898
 | title = Résultats et conjectures sur les nombres pratiques
 | volume = 299
 | year = 1984}}. As cited by {{harvtxt|Margenstern|1991}}.
*{{citation
 | last = Margenstern | first = Maurice
 | title = Les nombres pratiques: théorie, observations et conjectures
 | journal = [[Journal of Number Theory]]
 | volume = 37
 | year = 1991
 | issue = 1
 | pages = 1–36
 | mr = 1089787
 | doi = 10.1016/S0022-314X(05)80022-8| doi-access = free
 }}.
*{{citation
 | author-link = Giuseppe Melfi | last = Melfi | first = Giuseppe
 | title = A survey on practical numbers
 | journal = Rend. Sem. Mat. Univ. Pol. Torino
 | volume = 53
 | year = 1995
 | issue = 4
 | pages = 347–359
 }}.
*{{citation
 | last = Melfi | first = Giuseppe
 | title = On two conjectures about practical numbers
 | journal = Journal of Number Theory
 | volume = 56
 | year = 1996
 | issue = 1
 | pages = 205–210
 | mr = 1370203
 | doi = 10.1006/jnth.1996.0012| doi-access = free
 }}.
*{{citation
 | last1 = Mitrinović | first1 = Dragoslav S.
 | last2 = Sándor | first2 = József
 | last3 = Crstici | first3 = Borislav
 | contribution = III.50 Practical numbers
 | isbn = 978-0-7923-3823-9
 | pages = 118–119
 | publisher = Kluwer Academic Publishers
 | series = Mathematics and its Applications
 | title = Handbook of number theory, Volume 1
 | volume = 351
 | year = 1996}}.
*{{citation
 | last1 = Pomerance | first1 = C.
 | last2 = Weingartner | first2 = A.
 | journal = Ramanujan Journal
 | pages = 981–1000
 | title = On primes and practical numbers
 | volume = 57
 | issue = 3
 | year = 2021
 | doi = 10.1007/s11139-020-00354-y
 | arxiv = 2007.11062
 | s2cid = 220686445
 }}.
*{{citation
 | last = Robinson | first = D. F.
 | issue = 2
 | journal = New Zealand Math. Mag.
 | pages = 47–52
 | title = Egyptian fractions via Greek number theory
 | volume = 16
 | year = 1979}}. As cited by {{harvtxt|Margenstern|1991}} and {{harvtxt|Mitrinović|Sándor|Crstici|1996}}.
*{{citation
 | last = Saias | first = Eric
 | title = Entiers à diviseurs denses, I
 | journal = Journal of Number Theory
 | volume = 62
 | issue = 1
 | year = 1997
 | pages = 163–191
 | mr = 1430008
 | doi = 10.1006/jnth.1997.2057| doi-access = free
 }}.
*{{citation
 | author-link = Carlo Sanna | last = Sanna | first = Carlo
 | title = Practical numbers in Lucas sequences
 | journal = Quaestiones Mathematicae 
 | volume = 42
 | year = 2019
 | issue = 7
 | pages = 977–983
 | doi = 10.2989/16073606.2018.1502697
 | url = https://doi.org/10.2989/16073606.2018.1502697
 | hdl = 2318/1676275
 | hdl-access = free
 }}.
*{{citation
 | title = Fibonacci's Liber Abaci
 | last = Sigler | first = Laurence E. (trans.)
 | publisher = Springer-Verlag
 | year = 2002
 | isbn = 0-387-95419-8
 | pages = 119–121}}.
*{{citation
 | last = Sierpiński | first = Wacław | author-link = Wacław Sierpiński
 | doi = 10.1007/BF02410762
 | issue = 1
 | journal = Annali di Matematica Pura ed Applicata
 | pages = 69–74
 | title = Sur une propriété des nombres naturels
 | volume = 39
 | year = 1955| s2cid = 121592840 | doi-access = free
 }}.
*{{citation
 | last = Srinivasan | first = A. K.
 | title = Practical numbers
 | journal = [[Current Science]]
 | volume = 17
 | year = 1948
 | pages = 179–180
 | mr = 0027799
 | url = http://www.currentscience.ac.in/Downloads/article_id_017_06_0179_0180_0.pdf
 | archive-url = https://web.archive.org/web/20160305195057/http://www.currentscience.ac.in/Downloads/article_id_017_06_0179_0180_0.pdf
 | archive-date = 2016-03-05
 | url-status = dead}}.
*{{citation
 | last = Stewart | first = B. M. | author-link = Bonnie Stewart
 | title = Sums of distinct divisors
 | journal = [[American Journal of Mathematics]]
 | volume = 76
 | year = 1954
 | pages = 779–785
 | mr = 0064800
 | doi = 10.2307/2372651
 | jstor = 2372651
 | issue = 4
 | publisher = The Johns Hopkins University Press}}.
*{{citation
 | last1 = Tenenbaum | first1 = G. | author1-link = Gérald Tenenbaum
 | mr = 0860809
 | journal = Ann. Sci. Éc. Norm. Supér. |series=Série 4
 | pages = 1–30
 | issue = 1
 | title = Sur un problème de crible et ses applications
 | volume = 19
 | year = 1986| doi = 10.24033/asens.1502 | doi-access = free
 }}.
*{{citation
 | last1 = Tenenbaum | first1 = G. | author1-link = Gérald Tenenbaum
 | last2 = Yokota | first2 = H.
 | doi = 10.1016/0022-314X(90)90109-5
 | mr = 1057319
 | journal = Journal of Number Theory
 | pages = 150–156
 | issue = 2
 | title = Length and denominators of Egyptian fractions
 | volume = 35
 | year = 1990| doi-access = free
 }}.
*{{citation
 | last1 = Tenenbaum | first1 = G. | author1-link = Gérald Tenenbaum
 | last2 = Weingartner | first2 = A.
 | doi = 10.1093/qmath/haae002
 | mr = 4732950
 | journal = The Quarterly Journal of Mathematics
 | pages = 161–195
 | issue = 1
 | title = An Erdős-Kac theorem for integers with dense divisors
 | volume = 75
 | year = 2024
 | arxiv = 2211.05819
 }}.
*{{citation
 | last = Vose | first = M.
 | doi = 10.1112/blms/17.1.21
 | mr = 0766441
 | journal = [[London Mathematical Society|Bulletin of the London Mathematical Society]]
 | page = 21
 | title = Egyptian fractions
 | issue = 1
 | volume = 17
 | year = 1985}}.
*{{citation
 | last = Weingartner | first = A.
 | issue = 2
 | journal = The Quarterly Journal of Mathematics
 | pages = 743–758
 | title = Practical numbers and the distribution of divisors
 | volume = 66
 | year = 2015
 | doi = 10.1093/qmath/hav006| arxiv = 1405.2585
 }}.
*{{citation
 | last = Weingartner | first = A.
 | journal = Mathematics of Computation
 | pages = 1883–1902
 | title = On the constant factor in several related asymptotic estimates
 | volume = 88
 | issue = 318
 | year = 2019
 | doi = 10.1090/mcom/3402| arxiv = 1705.06349
 | s2cid = 85532616
 }}.
*{{citation
 | last = Weingartner | first = A.
 | journal = International Journal of Number Theory
 | pages = 629–638
 | title = The constant factor in the asymptotic for practical numbers
 | volume = 16
 | issue = 3
 | year = 2020
 | doi = 10.1142/S1793042120500311
 | arxiv = 1906.07819
 | s2cid = 195069356
 }}.
*{{citation
 | last = Weingartner | first = A.
 | journal = Proceedings of the American Mathematical Society
 | pages = 4699–4708
 | title = An extension of the Siegel-Walfisz theorem
 | volume = 149
 | issue = 11
 | year = 2021
 | doi = 10.1090/proc/15607
 | arxiv = 2011.06627
 | s2cid = 226956079
 }}.
*{{citation
 | last = Weingartner | first = A.
 | journal = Ramanujan Journal
 | pages = 447–453
 | title = Somewhat smooth numbers in short intervals
 | volume = 60
 | issue = 2
 | year = 2022
 | doi = 10.1007/s11139-022-00552-w
 | arxiv = 2105.13568
 | s2cid = 235247868
 }}.
*{{citation
 | last = Weingartner | first = A.
 | journal = International Journal of Number Theory
 | pages = 2333–2351
 | title = The mean number of divisors for rough, dense and practical numbers
 | volume = 19
 | issue = 10
 | year = 2023
 | doi = 10.1142/S1793042123501142
 | arxiv = 2104.07137
 }}.
{{refend}}