Golomb–Dickman constant

In mathematics, the Golomb–Dickman constant, named after Solomon W. Golomb and Karl Dickman, is a mathematical constant, which arises in the theory of random permutations and in number theory. Its value is

\lambda =0.62432998854355087099293638310083724\dots

(sequence A084945 in the OEIS)

It is not known whether this constant is rational or irrational.^[1]

It's simple continued fraction is given by $[0;1,1,1,1,1,22,1,2,3,1,...]$ , which appears to have an unusually large number of 1s.^[2]

Definitions

Let a_n be the average — taken over all permutations of a set of size n — of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is

\lambda =\lim _{n\to \infty }{\frac {a_{n}}{n}}.

In the language of probability theory, $\lambda n$ is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size n.

In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer. More precisely,

\lambda =\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=2}^{n}{\frac {\log(P_{1}(k))}{\log(k)}},

where $P_{1}(k)$ is the largest prime factor of k (sequence A006530 in the OEIS) . So if k is a d digit integer, then $\lambda d$ is the asymptotic average number of digits of the largest prime factor of k.

The Golomb–Dickman constant appears in number theory in a different way. What is the probability that second largest prime factor of n is smaller than the square root of the largest prime factor of n? Asymptotically, this probability is $\lambda$ . More precisely,

\lambda =\lim _{n\to \infty }{\text{Prob}}\left\{P_{2}(n)\leq {\sqrt {P_{1}(n)}}\right\}

where $P_{2}(n)$ is the second largest prime factor n.

The Golomb-Dickman constant also arises when we consider the average length of the largest cycle of any function from a finite set to itself. If X is a finite set, if we repeatedly apply a function f: X → X to any element x of this set, it eventually enters a cycle, meaning that for some k we have $f^{n+k}(x)=f^{n}(x)$ for sufficiently large n; the smallest k with this property is the length of the cycle. Let b_n be the average, taken over all functions from a set of size n to itself, of the length of the largest cycle. Then Purdom and Williams^[3] proved that

\lim _{n\to \infty }{\frac {b_{n}}{\sqrt {n}}}={\sqrt {\frac {\pi }{2}}}\lambda .

Formulae

There are several expressions for $\lambda$ . These include:^[4]

\lambda =\int _{0}^{1}e^{\mathrm {li} (t)}\,dt

where $\mathrm {li} (t)$ is the logarithmic integral,

\lambda =\int _{0}^{\infty }e^{-t-E_{1}(t)}\,dt

where $E_{1}(t)$ is the exponential integral, and

\lambda =\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}\,dt

and

\lambda =\int _{0}^{\infty }{\frac {\rho (t)}{(t+1)^{2}}}\,dt

where $\rho (t)$ is the Dickman function.

External links

Weisstein, Eric W. "Golomb-Dickman Constant". MathWorld.
OEIS sequence A084945 (Decimal expansion of Golomb-Dickman constant)
Finch, Steven R. (2003). Mathematical Constants. Cambridge University Press. pp. 284–286. ISBN 0-521-81805-2.

References

^ Lagarias, Jeffrey (2013). "Euler's constant: Euler's work and modern developments". Bull. Amer. Math. Soc. 50 (4): 527–628. arXiv:1303.1856. Bibcode:2013arXiv1303.1856L. doi:10.1090/S0273-0979-2013-01423-X. S2CID 119612431.
^ Weisstein, Eric W. "Golomb-Dickman Constant Continued Fraction". mathworld.wolfram.com. Retrieved 2024-10-11.
^ Purdon, P.; Williams, J.H (1968). "Cycle length in a random function". Trans. Amer. Math. Soc. 133 (2): 547–551. doi:10.1090/S0002-9947-1968-0228032-3.
^ Weisstein, Eric W. "Golomb-Dickman Constant". mathworld.wolfram.com. Retrieved 2024-10-11.

Read other articles:

List of ancient Armeno-Phrygian peoples and tribes

Part of a series onIndo-European topics Languages List of Indo-European languages Extant Albanian Armenian Balto-Slavic Baltic Slavic Celtic Germanic Hellenic Greek Kurdish Indo-Iranian Indo-Aryan Iranian Nuristani Italic Romance Extinct Anatolian Tocharian Paleo-Balkan Dacian Illyrian Liburnian Messapian Mysian Paeonian Phrygian Thracian Reconstructed Proto-Indo-European language Phonology: Sound laws, Accent, Ablaut Hypothetical Daco-Thracian Graeco-Armenian Graeco-Aryan Graeco-Phrygian In...

Equisetopsida

Equisetopsida Equisetum fluviatile (en) TaksonomiSuperkerajaanEukaryotaKerajaanPlantaeDivisiEquisetophytaKelasEquisetopsida C.Agardh, 1825 Tata namaSinonim taksonSphenopsidaOrdo Equisetales † Archaeocalamitaceae † Calamitaceae Equisetaceae † Pseudoborniales † Sphenophyllales lbs Equisetopsida, atau Sphenopsida di kalangan paleobotani, adalah salah satu kelas tumbuhan berpembuluh yang masih lestari hingga kini namun dengan temuan fosil berusia sampai ke kala Devon. ...

Воинское звание

У этого термина существуют и другие значения, см. Звание. Не следует путать со специальным званием. Запрос «Корабельное воинское звание» перенаправляется сюда. На эту тему нужно создать отдельную статью. Царский стольник в книге «Одежды Русского государства» Фёдора Сол�...

1 Korintus 16

1 Korintus 16Surat 1 Korintus 7:33-8:4 yang tertulis pada naskah Papirus 15, dibuat sekitar abad ke-3 M.KitabSurat 1 KorintusKategoriSurat-surat PaulusBagian Alkitab KristenPerjanjian BaruUrutan dalamKitab Kristen7← pasal 15 2 Korintus 1 → 1 Korintus 16 (atau I Korintus 16, disingkat 1Kor 16) adalah bagian terakhir surat rasul Paulus yang pertama kepada jemaat di Korintus dalam Perjanjian Baru di Alkitab Kristen.[1][2] Dibuat oleh rasul Paulus dan Sostenes[3 ...

شبكة حاسوب

أنظمة التشغيل الميزّات المشتركة إدارة عملية مقاطعات تنظيم الذاكرة نظام الملف مشغلات الأجهزة التشبيك الأمن واجهة المستخدم الرسومية عنت شبكة حاسوب علم الشبكات نظرية الشبكات الرسم البيانيّ تعقيد الشبكة شبكات العالم الصغير الشبكات غير المحدودة بينة المجتمع نظرية التخلل ال...

J. R. Greville

John Rodger Greville John Rodger Greville (15 June 1834 – 29 April 1894)[1] was an Irish-born comic actor, singer, songwriter and stage manager who had a long career in Australia. History Greville was born John Rodger in Dublin, Ireland, a son of John Rodger JP (c. 1812–1897) and his wife Jane Rodger, née Greville (died 1899). His first acting experience was in May 1851, when he played Rochester in a comedy entitled Charles II, with Dublin's Royal Phoenix Amateur Theatre.[2&...

Miracle in Motown

Notable American football play Miracle in MotownFord Field, the site of the game. Green Bay Packers (7–4) Detroit Lions (4–7) 27 23 Head coach:Mike McCarthy Head coach:Jim Caldwell 1234 Total GB 001413 27 DET 17033 23 DateDecember 3, 2015StadiumFord Field, Detroit, MichiganRefereeCarl CheffersTV in the United StatesNetworkCBS, NFL NetworkAnnouncersJim Nantz, Phil Simms, Tracy Wolfson The Miracle in Motown was the final play of an American football game between the NFC North divisional riv...

Sepeda motor

Artikel ini bukan mengenai Motor. Sepeda motor Norton. Sepeda motor adalah kendaraan beroda dua[1] yang digerakkan oleh sebuah mesin. Letak kedua roda sebaris lurus dan pada kecepatan tinggi sepeda motor tetap stabil disebabkan oleh gaya giroskopik. Sedangkan pada kecepatan rendah, kestabilan atau keseimbangan sepeda motor bergantung kepada pengaturan setang oleh pengendara. Penggunaan sepeda motor di Indonesia sangat populer karena harganya yang relatif murah, terjangkau untuk sebagi...

Menara Maxis

Menara Maxis merupakan menara ketujuh tertinggi di Malaysia. Menara ini terletak bersebelahan dengan Menara PETRONAS, Kuala Lumpur. Menara ini adalah setinggi 212 meter tinggi. Di sini terdapat ibu pejabat Maxis Communications dan Tanjong Plc Group of Companies. Di bagian atas terdapat sebuah jam digital yang besar bagi kegunaan untuk melihat waktu. Menara Maxis. Pranala luar Halaman web Maxis Diarsipkan 2008-04-20 di Wayback Machine. Artikel mengenai Menara Maxis di MyPageNation Menara Maxis...

Синелобый амазон

Синелобый амазон Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеНадкласс:ЧетвероногиеКлада:АмниотыКлада:ЗавропсидыКласс:Пт�...

Itō Sukeyuki

CountItō Sukeyuki 伊東祐亨Nama asli伊東祐亨Lahir(1843-05-20)20 Mei 1843Kagoshima, Domain Satsuma, JepangMeninggal16 Januari 1914(1914-01-16) (umur 70)[1]Tokyo, JepangPengabdian Kekaisaran JepangDinas/cabang Angkatan Laut Kekaisaran JepangLama dinas1871–1907Pangkat Marsekal LaksamanaKomandan Daiichi Teibo Kasuga Azuma Nisshin Ryūjō Hiei Fusō Naniwa Armada Gabungan Distrik Angkatan Laut Yokosuka Kepala Staf Umum IJN Perang/pertempuran Perang Inggris-Satsu...

Booting process of Android devices

Boot for Android devices The booting process of Android devices starts at the power-on of the SoC (system on a chip) and ends at the visibility of the home screen, or special modes like recovery and fastboot.[a] The boot process of devices that run Android is influenced by the firmware design of the SoC manufacturers. Background As of 2018, 90% of the SoCs of the Android market are supplied by either Qualcomm, Samsung or MediaTek.[1] Other vendors include Rockchip, Marvell, Nv...

约翰斯顿环礁

此條目可参照英語維基百科相應條目来扩充。 (2021年5月6日)若您熟悉来源语言和主题，请协助参考外语维基百科扩充条目。请勿直接提交机械翻译，也不要翻译不可靠、低品质内容。依版权协议，译文需在编辑摘要注明来源，或于讨论页顶部标记{{Translated page}}标签。约翰斯顿环礁Kalama Atoll 美國本土外小島嶼 Johnston Atoll 旗幟颂歌：《星條旗》The Star-Spangled Banner約翰斯頓環礁�...

Feres, Evros

For the municipality in the Magnesia prefecture, see Feres, Magnesia. Municipal unit in GreeceFeres ΦέρεςMunicipal unitFeresLocation within the regional unit Coordinates: 40°53′30″N 26°10′15″E / 40.89167°N 26.17083°E / 40.89167; 26.17083CountryGreeceAdministrative regionEast Macedonia and ThraceRegional unitEvrosMunicipalityAlexandroupoliMunicipality established1986Area • Municipal unit411.2 km2 (158.8 sq mi)Elevation43 ...

Laksamana Pertama

Pangkat militer Indonesia Angkatan Darat Angkatan Laut Angkatan Udara Perwira Jenderal Besar Laksamana Besar Marsekal Besar Jenderal Laksamana Marsekal Letnan Jenderal Laksamana Madya Marsekal Madya Mayor Jenderal Laksamana Muda Marsekal Muda Brigadir Jenderal Laksamana Pertama Marsekal Pertama Kolonel Kolonel Kolonel Letnan Kolonel Letnan Kolonel Letnan Kolonel Mayor Mayor Mayor Kapten Kapten Kapten Letnan Satu Letnan Satu Letnan Satu Letnan Dua Letnan Dua Letnan Dua Bintara Pembantu Letnan...

Pulau Buka

Pulau Buka dari ruang angkasa Pulau Buka adalah pulau terbesar kedua di Wilayah Otonomi Bougainville, timur Papua Nugini. Pulau tersebut berada di Distrik Bougainville Utara, dengan ibu kota distrik dan Wilayah Otonomi Buka di pulau tersebut. Pulau tersebut dipisahkan dari pantai barat laut Pulau Bougainville oleh selat Buka. Referensi Blackwood, B. 1933 Both Sides of Buka Passage. Oxford: Clarendon Press. Reagan A.J and Griffin, J.M (eds) 2005 Bougainville Before the Conflict. Canberra: Pand...

President of Russia

Since 1991, head of state of the RSFSR and Russia President of the Russian FederationПрезидент Российской ФедерацииPresidential emblemPresidential standardIncumbentVladimir Putinsince 7 May 2012Presidential Administration of RussiaStyleMr President(informal)Comrade Supreme Commander(military)His Excellency[1](diplomatic)TypePresidentStatusHead of state Commander-in-chiefMember ofState CouncilSecurity CouncilSupreme Eurasian Economic CouncilResidenceMos...

Sebastian Coe

British athlete and politician (born 1956) The Right HonourableThe Lord CoeCH KBE Hon FRIBACoe during the World Economic Forum in Davos, Switzerland, in 2012President of World AthleticsIncumbentAssumed office 19 August 2015Preceded byLamine DiackChairman of the British Olympic AssociationIn office7 November 2012 – 24 November 2016PresidentThe Princess RoyalPreceded byThe Lord MoynihanSucceeded bySir Hugh RobertsonChairman of the London Organising Committee of the ...

Daftar keuskupan di Amerika Serikat

Daftar keuskupan di Amerika Serikat adalah sebuah daftar yang memuat dan menjabarkan pembagian terhadap wilayah administratif Gereja Katolik Roma yang dipimpin oleh seorang uskup ataupun ordinaris di Amerika Serikat. Konferensi para uskup Amerika Serikat bergabung dalam Konferensi Waligereja Amerika Serikat (USCCB). Per Mei 2020, terdapat 33 buah keuskupan agung Ritus Roma, 1 buah metropolitan Ritus Rutenia, dan 159 keuskupan sufragan (18 di antaranya merupakan Ritus Timur), 1 ordinariat mili...

Granary Square

Urban square in the London Borough of Camden Road bridge to Granary Square with Central Saint Martins college behind Granary Square is a large open space in the London Borough of Camden. Comparable in size to Trafalgar Square,[1] it is part of the larger King's Cross Central development. It has been cited as an example of a privately owned public space in London.[2] The new Central Saint Martins complex is adjacent. Visitor attractions in the square include Queer Britain, the ...

/profillengkap.com/index.php/article/Dafo Temple (Xinchang)

/profillengkap.com/index.php/article/Is This the Life We Really Want?

/profillengkap.com/index.php/article/State symbols of the President of Ukraine

/profillengkap.com/index.php/article/Isaac Roach

Demografi Serbia

Calcio Fiorentino

Daftar pondok pesantren di Sulawesi Barat

Cal 9.2

Girija Devi (politician)

بوابة:جنوب السودان

El Real de la Jara

1985 Philadelphia Phillies season