Unitary perfect number

Unsolved problem in mathematics:
Are there infinitely many unitary perfect numbers?
(more unsolved problems in mathematics)

A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor d of a number n is a unitary divisor if d and n/d share no common factors). The number 6 is the only number that is both a perfect number and a unitary perfect number.

Known examples

The number 60 is a unitary perfect number because 1, 3, 4, 5, 12, 15, and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first five, and only known, unitary perfect numbers are:

  • , and
  • (sequence A002827 in the OEIS).

The respective sums of their proper unitary divisors are as follows:

  • 6 = 1 + 2 + 3
  • 60 = 1 + 3 + 4 + 5 + 12 + 15 + 20
  • 90 = 1 + 2 + 5 + 9 + 10 + 18 + 45
  • 87360 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 64 + 65 + 91 + 105 + 192 + 195 + 273 + 320 + 448 + 455 + 832 + 960 + 1344 + 1365 + 2240 + 2496 + 4160 + 5824 + 6720 + 12480 + 17472 + 29120
  • 146361946186458562560000 = 1 + 3 + 7 + 11 + ... + 13305631471496232960000 + 20908849455208366080000 + 48787315395486187520000 (4095 divisors in the sum)

Properties

There are no odd unitary perfect numbers. This follows since 2d*(n) divides the sum of the unitary divisors of an odd number n, where d*(n) is the number of distinct prime factors of n. One gets this because the sum of all the unitary divisors is a multiplicative function and one has that the sum of the unitary divisors of a prime power pa is pa + 1 which is even for all odd primes p. Therefore, an odd unitary perfect number must have only one distinct prime factor, and it is not hard to show that a power of prime cannot be a unitary perfect number, since there are not enough divisors.

It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known. A sixth such number would have at least nine distinct odd prime factors.[1]

References

  1. ^ Wall, Charles R. (1988). "New unitary perfect numbers have at least nine odd components". Fibonacci Quarterly. 26 (4): 312–317. doi:10.1080/00150517.1988.12429611. ISSN 0015-0517. MR 0967649. Zbl 0657.10003.

Read other articles:

Stasiun Kaji

Stasiun Kaji加治駅Stasiun Kaji pada Agustus 2004Lokasi165 Shimonaka, Shibata-shi, Niigata-ken 959-2451JepangKoordinat37°58′37.3″N 139°21′32.5″E / 37.977028°N 139.359028°E / 37.977028; 139.359028Koordinat: 37°58′37.3″N 139°21′32.5″E / 37.977028°N 139.359028°E / 37.977028; 139.359028Operator JR EastJalur■ Jalur Utama UetsuLetak30.3 km dari NiitsuJumlah peron1 peron pulauJumlah jalur2Informasi lainStatusTanpa stafSitus ...

 

Bukan Pocong Biasa

Bukan Pocong BiasaSutradaraChiska DoppertProduserFirman BintangDitulis olehTb Ule Sulaeman SPemeran Zacky Zimah Soraya Larasati Ronald Gustav Yoga Arizona Irene Justine Mucle Malih Tong Tong Mpok Atiek Randy Pangalila Perusahaanproduksi Mitra Pictures Bic Productions Tanggal rilis 22 Desember 2011 (2011-12-22) Durasi85 menitNegaraIndonesiaBahasaBahasa Indonesia Bukan Pocong Biasa merupakan film horor/komedi Indonesia yang dirilis pada 22 Desember 2011 yang disutradarai oleh Chiska Dopper...

 

Ginjal

GinjalGinjal dilihat dari belakang (tulang rusuk dihilangkan).RincianSistemSistem perkemihan dan sistem endokrinArteriArteri renalisVenaVena renalisSarafPleksus renalisPengidentifikasiBahasa LatinRenYunaniNephrosMeSHD007668TA98A08.1.01.001TA23358FMA7203Daftar istilah anatomi[sunting di Wikidata] Ginjal atau buah pinggang adalah organ ekskresi dalam vertebrata yang berbentuk mirip kacang. Dalam manusia dewasa, ukuran ginjal sekitar 11 sentimeter panjangnya. Ginjal menerima darah dari sepas...

العلاقات الصينية المكسيكية

العلاقات الصينية المكسيكية الصين المكسيك   الصين   المكسيك تعديل مصدري - تعديل   العلاقات الصينية المكسيكية هي العلاقات الثنائية التي تجمع بين الصين والمكسيك.[1][2][3][4][5] مقارنة بين البلدين هذه مقارنة عامة ومرجعية للدولتين: وجه المقارنة الص�...

 

Captaincy General of Cuba

1607–1898 Spanish possession in the Caribbean This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Captaincy General of Cuba – news · newspapers · books · scholar · JSTOR (May 2009) (Learn how and when to remove this template message) Captaincy General of CubaCapitanía General de Cuba (Spanish)1607–1899...

 

1940 United States presidential election in Arizona

Main article: 1940 United States presidential election 1940 United States presidential election in Arizona ← 1936 November 5, 1940[1] 1944 → All 3 Arizona votes to the Electoral College   Nominee Franklin D. Roosevelt Wendell Willkie Party Democratic Republican Home state New York New York Running mate Henry A. Wallace Charles L. McNary Electoral vote 3 0 Popular vote 95,267 54,030 Percentage 63.49% 36.01% County Results Roosevelt &...

James McClean

James McClean Informasi pribadiNama lengkap James Aaron McCleanTanggal lahir 22 April 1989 (umur 34)Tempat lahir Derry, Irlandia UtaraTinggi 1,80 m (5 ft 11 in)Posisi bermain Gelandang SayapInformasi klubKlub saat ini West Bromwich AlbionNomor 14Karier junior Trojans InstituteKarier senior*Tahun Tim Tampil (Gol)2007 Institute 1 (0)2008–2011 Derry City 73 (18)2011–2013 Sunderland 70 (11)2013–2015 Wigan Athletic 73 (9)2015– West Bromwich Albion 68 (3)Tim nasional‡...

 

Ash Ketchum

Ash KetchumTokoh PokémonKarya seni Ash Ketchum yang dimunculkan dalam anime musim ke-23 dan ke-24.PenampilanperdanaPokémon, I Choose You! (1997)PenampilanterakhirThe Rainbow and the Pokémon Master! (2023)PenciptaSatoshi TajiriDidasarkandariRedDidesainolehAtsuko NishidaPengisi suaraBahasa Jepang:Rica Matsumoto[1]Bahasa Inggris:Veronica Taylor (musim 1–8)[2]Sarah Natochenny (musim 9–25)[3]InformasiJenis kelaminLaki-lakiPekerjaanPelatih PokémonKeluargaDelia Ketchu...

 

George Dvorsky

Canadian bioethicist, transhumanist, and futurist This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) The topic of this article may not meet Wikipedia's notability guideline for biographies. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notab...

Tom Wilkinson

Tom WilkinsonOBEWilkinson pada 2009LahirThomas Geoffrey Wilkinson(1948-02-05)5 Februari 1948Leeds, Yorkshire, InggrisMeninggal30 Desember 2023(2023-12-30) (umur 75)London, InggrisPendidikanUniversitas KentAkademi Seni Drama KerajaanPekerjaanAktorTahun aktif1976–2023Suami/istriDiana Hardcastle ​(m. 1988)​Anak2 Thomas Geoffrey Wilkinson OBE (5 Februari 1948 – 30 Desember 2023) merupakan seorang aktor berkebangsaan Inggris yang memenangkan nominasi Aca...

 

Gilze en Rijen

Gilze en RijenKota BenderaLambang kebesaranNegaraBelandaProvinsiBrabant UtaraLuas(2006) • Total65,67 km2 (2,536 sq mi) • Luas daratan65,54 km2 (2,531 sq mi) • Luas perairan0,13 km2 (5 sq mi)Populasi (1 Januari 2007) • Total25.381 • Kepadatan387/km2 (1,000/sq mi) Sumber: CBS, Statline.Zona waktuUTC+1 (CET) • Musim panas (DST)UTC+2 (CEST) Gilze en Rijen, adalah seb...

 

Tila Tequila

American media personality (born 1981) Paralyze (song) redirects here. Not to be confused with Paralyzer. Tila TequilaTequila in October 2008BornThien Thanh Thi Nguyen (1981-10-24) October 24, 1981 (age 42)SingaporeNationalityAmericanOther namesTila NguyenMiss TilaTornado ThienOccupations Model television personality singer Years active2001–presentChildren2Musical careerGenres Hip hop pop rap pop rock electropop Labels The Saturday Team will.i.am Music Group Musical artist Ng...

土库曼斯坦总统

土库曼斯坦总统土库曼斯坦国徽土库曼斯坦总统旗現任谢尔达尔·别尔德穆哈梅多夫自2022年3月19日官邸阿什哈巴德总统府（Oguzkhan Presidential Palace）機關所在地阿什哈巴德任命者直接选举任期7年，可连选连任首任萨帕尔穆拉特·尼亚佐夫设立1991年10月27日 土库曼斯坦土库曼斯坦政府与政治 国家政府 土库曼斯坦宪法 国旗 国徽 国歌 立法機關（英语：National Council of Turkmenistan） ...

 

Calvin Smith

American sprinter For other people named Calvin Smith, see Calvin Smith (disambiguation). Calvin SmithPersonal informationNationalityAmericanBorn (1961-01-08) January 8, 1961 (age 63)Bolton, MississippiHeight5 ft 10 in (1.78 m)Weight152 lb (69 kg)SportSportTrack and fieldEventSprintsCollege teamAlabama Crimson TideAchievements and titlesPersonal bests100 m: 9.93 s (Colorado Springs, 1983)200 m: 19.99 s (Zürich, 1983) Medal record Men's athletics Repres...

 

Super Mario World (TV series)

Television series For other uses, see Super Mario World (disambiguation). This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Super Mario World TV series – news · newspapers · books · scholar · JSTOR (October 2008) (Learn how and when to remove this message) Super Mario WorldTitle cardGenre Action Adventure...

إذاعة البرنامج الموسيقي

هذه المقالة تحتاج للمزيد من الوصلات للمقالات الأخرى للمساعدة في ترابط مقالات الموسوعة. فضلًا ساعد في تحسين هذه المقالة بإضافة وصلات إلى المقالات المتعلقة بها الموجودة في النص الحالي. (ديسمبر 2018) يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقال�...

 

Battle of Ridgeway

Part of the Fenian raids Battle of RidgewayPart of the Fenian RaidsAn 1869 illustration of the battle: Charge of General O'Neill's Fenians upon the Canadian troops, causing their rout.Date2 June 1866LocationRidgeway, Canada West42°54′16″N 79°02′30″W / 42.90444°N 79.04167°W / 42.90444; -79.04167Result Fenian victoryBelligerents Fenian Brotherhood Province of CanadaCommanders and leaders Brig. Gen. John O'Neill Lt. Col. Alfred BookerStrength 600–700[1&#...

 

Sectarianism

Conflict between groups related to the form of government they live under Not to be confused with Secularism. This article is about sectarianism and its different conceptual interpretations. For sectarianism as a characteristic of sects, see sect. Part of a series onDiscrimination Forms Institutional Structural Statistical Taste-based Attributes Age Caste Class Dialect Disability Genetic Hair texture Height Language Looks Mental disorder Race / Ethnicity Skin color Scientific racism ...

Naskapi

See also: Naskapi language Ethnic group of Quebec and Labrador, Canada This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Naskapi – news · newspapers · books · scholar · JSTOR (January 2009) (Learn how and when to remove this message) Ethnic group Naskapi(ᓇᔅᑲᐱ) Flag of the Kawawachikamach Band of the Na...

 

Northern German Football Association

Regional association of the German Football Association Northern German Football AssociationFormation19051948 (reformed)TypeFootball associationLocationBremen, GermanyMembership 1,052,406 (2017)PresidentEugen GehlenborgParent organizationGerman Football AssociationAffiliationsBremen Football AssociationHamburg Football AssociationLower Saxony Football AssociationSchleswig-Holstein Football AssociationWebsitewww.nordfv.de The Northern German Football Association (German: Norddeutscher Fußball...