The tables contain the prime factorization of the natural numbers from 1 to 1000.
When n is a prime number, the prime factorization is just n itself, written in bold below.
The number 1 is called a unit. It has no prime factors and is neither prime nor composite.
Properties Many properties of a natural number can be seen or directly computed from the prime factorization of . The multiplicity of a prime factor of is the largest exponent for which divides . The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is (since ). The multiplicity of a prime which does not divide may be called or may be considered undefined. and , the prime omega functions, count the number of prime factors of a natural number . (little omega) is the number of distinct prime factors of . (big omega) is the number of prime factors of counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has . The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 . There are many special types of prime numbers. A composite number has . The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 . All numbers above 1 are either prime or composite. 1 is neither. A semiprime has (so it is composite). The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34 . A -almost prime (for a natural number ) has (so it is composite if ). An even number has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 . An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 . All integers are either even or odd. A square has even multiplicity for all prime factors (it is of the form for some ). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 . A cube has all multiplicities divisible by 3 (it is of the form for some ). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 . A perfect power has a common divisor for all multiplicities (it is of the form for some and ). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 . 1 is sometimes included. A powerful number (also called squarefull) has multiplicity greater than 1 for all its prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72 . A prime power has only one prime factor, i.e. . The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 . 1 is sometimes included. An Achilles number is powerful but not a perfect power. The first: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968 . A square-free integer has no prime factor with multiplicity greater than 1. The first: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17 . A number where some but not all prime factors have multiplicity greater than 1 is neither square-free nor squarefull, but squareful. The Liouville function is 1 if is even, and is -1 if is odd. The Möbius function is 0 if is not square-free. Otherwise is 1 if is even, and is −1 if is odd. A sphenic number is square-free and the product of 3 distinct primes, i.e. it has . The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 . , sometimes called the integer logarithm, is the sum of primes dividing , counted with multiplicity. It is an additive function. A Ruth-Aaron pair is a pair of two consecutive numbers with . The first (by value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 . Another definition is where the same prime is only counted once; if so, the first (by value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 . A primorial is the product of all primes from 2 to . The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 . is sometimes included. A factorial is the product of all numbers from 1 to . The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 . is sometimes included. A -smooth number (for a natural number ) has its prime factors (so it is also -smooth for any ). is smoother than if the largest prime factor of is less than the largest of . A regular number has no prime factor greater than 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 . A -powersmooth number has all where is a prime factor with multiplicity . A frugal number has more digits than the number of digits in its prime factorization (when written like the tables below with multiplicities above 1 as exponents). The first in decimal: 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250 . An equidigital number has the same number of digits as its prime factorization. The first in decimal: 1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17 . An extravagant number has fewer digits than its prime factorization. The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 . An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital. (greatest common divisor of and ) is the product of all prime factors which are both in and (with the smallest multiplicity for and ). and are coprime (also called relatively prime) if they have no common prime factors, which implies . (least common multiple of and ) is the product of all prime factors of or (with the largest multiplicity for or ). . Finding the prime factors is often harder than computing and using other algorithms which do not require known prime factorization. is a divisor of (also called divides , or is divisible by ) if all prime factors of have at least the same multiplicity in . The divisors of are all products of some or all prime factors of (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown in table of divisors.
1 to 100 + 1–201 22334225562·377823932102·511111222·31313142·7153·516241717182·3219192022·5+ 21–40213·7222·1123232423·32552262·1327332822·72929302·3·531313225333·11342·17355·73622·323737382·19393·134023·5+ 41–604141422·3·743434422·114532·5462·2347474824·34972502·52513·175222·135353542·33555·115623·7573·19582·2959596022·3·5+ 61–806161622·316332·76426655·13662·3·1167676822·17693·23702·5·771717223·327373742·37753·527622·19777·11782·3·1379798024·5+ 81–1008134822·4183838422·3·7855·17862·43873·298823·118989902·32·5917·139222·23933·31942·47955·199625·39797982·729932·1110022·52
101 to 200 + 101–1201011011022·3·1710310310423·131053·5·71062·5310710710822·331091091102·5·111113·3711224·71131131142·3·191155·2311622·2911732·131182·591197·1712023·3·5+ 121–1401211121222·611233·4112422·31125531262·32·7127127128271293·431302·5·1313113113222·3·111337·191342·6713533·513623·171371371382·3·2313913914022·5·7+ 141–1601413·471422·7114311·1314424·321455·291462·731473·7214822·371491491502·3·5215115115223·1915332·171542·7·111555·3115622·3·131571571582·791593·5316025·5+ 161–1801617·231622·3416316316422·411653·5·111662·8316716716823·3·71691321702·5·1717132·1917222·431731731742·3·2917552·717624·111773·591782·8917917918022·32·5+ 181–2001811811822·7·131833·6118423·231855·371862·3·3118711·1718822·4718933·71902·5·1919119119226·31931931942·971953·5·1319622·721971971982·32·1119919920023·52
201 to 300 + 201–2202013·672022·1012037·2920422·3·172055·412062·10320732·2320824·1320911·192102·3·5·721121121222·532133·712142·1072155·4321623·332177·312182·1092193·7322022·5·11+ 221–24022113·172222·3·3722322322425·722532·522262·11322722722822·3·192292292302·5·232313·7·1123223·292332332342·32·132355·4723622·592373·792382·7·1723923924024·3·5+ 241–2602412412422·1122433524422·612455·722462·3·4124713·1924823·312493·832502·5325125125222·32·725311·232542·1272553·5·17256282572572582·3·432597·3726022·5·13+ 261–28026132·292622·13126326326423·3·112655·532662·7·192673·8926822·672692692702·33·527127127224·172733·7·132742·13727552·1127622·3·232772772782·13927932·3128023·5·7+ 281–3002812812822·3·4728328328422·712853·5·192862·11·132877·4128825·322891722902·5·292913·9729222·732932932942·3·722955·5929623·3729733·112982·14929913·2330022·3·52
301 to 400 + 301–3203017·433022·1513033·10130424·193055·613062·32·1730730730822·7·113093·1033102·5·3131131131223·3·133133133142·15731532·5·731622·793173173182·3·5331911·2932026·5+ 321–3403213·1073222·7·2332317·1932422·3432552·133262·1633273·10932823·413297·473302·3·5·1133133133222·8333332·373342·1673355·6733624·3·73373373382·1323393·11334022·5·17+ 341–36034111·313422·32·193437334423·433453·5·233462·17334734734822·3·293493493502·52·735133·1335225·113533533542·3·593555·7135622·893573·7·173582·17935935936023·32·5+ 361–3803611923622·1813633·11236422·7·133655·733662·3·6136736736824·2336932·413702·5·373717·5337222·3·313733733742·11·173753·5337623·4737713·293782·33·737937938022·5·19+ 381–4003813·1273822·19138338338427·33855·7·113862·19338732·4338822·973893893902·3·5·1339117·2339223·723933·1313942·1973955·7939622·32·113973973982·1993993·7·1940024·52
401 to 500 + 401–4204014014022·3·6740313·3140422·10140534·54062·7·2940711·3740823·3·174094094102·5·414113·13741222·1034137·594142·32·234155·8341625·134173·1394182·11·1941941942022·3·5·7+ 421–4404214214222·21142332·4742423·5342552·174262·3·714277·6142822·1074293·11·134302·5·4343143143224·334334334342·7·314353·5·2943622·10943719·234382·3·7343943944023·5·11+ 441–46044132·724422·13·1744344344422·3·374455·894462·2234473·14944826·74494494502·32·5245111·4145222·1134533·1514542·2274555·7·1345623·3·194574574582·22945933·1746022·5·23+ 461–4804614614622·3·7·1146346346424·294653·5·314662·23346746746822·32·134697·674702·5·474713·15747223·5947311·434742·3·7947552·1947622·7·1747732·534782·23947947948025·3·5+ 481–50048113·374822·2414833·7·2348422·1124855·974862·3548748748823·614893·1634902·5·7249149149222·3·4149317·294942·13·1949532·5·1149624·314977·714982·3·8349949950022·53
501 to 600 + 501–5205013·1675022·25150350350423·32·75055·1015062·11·235073·13250822·1275095095102·3·5·175117·735122951333·195142·2575155·10351622·3·4351711·475182·7·375193·17352023·5·13+ 521–5405215215222·32·2952352352422·1315253·52·75262·26352717·3152824·3·115292325302·5·5353132·5953222·7·1953313·415342·3·895355·10753623·675373·1795382·26953972·1154022·33·5+ 541–5605415415422·2715433·18154425·175455·1095462·3·7·1354754754822·13754932·615502·52·1155119·2955223·3·235537·795542·2775553·5·3755622·1395575575582·32·3155913·4356024·5·7+ 561–5805613·11·175622·28156356356422·3·475655·1135662·28356734·756823·715695695702·3·5·1957157157222·11·135733·1915742·7·4157552·2357626·325775775782·1725793·19358022·5·29+ 581–6005817·835822·3·9758311·5358423·7358532·5·135862·29358758758822·3·7258919·315902·5·595913·19759224·375935935942·33·115955·7·1759622·1495973·1995982·13·2359959960023·3·52
601 to 700 + 601–6206016016022·7·4360332·6760422·1516055·1126062·3·10160760760825·196093·7·296102·5·6161113·4761222·32·176136136142·3076153·5·4161623·7·116176176182·3·10361961962022·5·31+ 621–64062133·236222·3116237·8962424·3·13625546262·3136273·11·1962822·15762917·376302·32·5·763163163223·796333·2116342·3176355·12763622·3·5363772·136382·11·2963932·7164027·5+ 641–6606416416422·3·10764364364422·7·236453·5·436462·17·1964764764823·3464911·596502·52·136513·7·3165222·1636536536542·3·1096555·13165624·4165732·736582·7·4765965966022·3·5·11+ 661–6806616616622·3316633·13·1766423·836655·7·196662·32·3766723·2966822·1676693·2236702·5·6767111·6167225·3·76736736742·33767533·5267622·1326776776782·3·1136797·9768023·5·17+ 681–7006813·2276822·11·3168368368422·32·196855·1376862·736873·22968824·4368913·536902·3·5·2369169169222·17369332·7·116942·3476955·13969623·3·2969717·416982·3496993·23370022·52·7
701 to 800