Natural number
Cardinal seventy-two Ordinal 72nd Factorization 23 × 32 Divisors 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 (12) Greek numeral ΟΒ´ Roman numeral LXXII ,lxxii Binary 10010002 Ternary 22003 Senary 2006 Octal 1108 Duodecimal 6012 Hexadecimal 4816
72 (seventy-two ) is thenatural number following71 and preceding73 . It is half agross and also sixdozen (i.e., 60 induodecimal ).
Seventy-two is apronic number , as it is the product of8 and9 .[ 1] Achilles number , as it is apowerful number that is not itself apower .[ 2]
72 is anabundant number .[ 3] 12 (one of only twosublime numbers ),[ 4] refactorable numbers .[ 5] largely composite number .[ 6] Euler totient of24 .[ 7] highly totient number , as there are17 solutions to the equation φ(x ) = 72, more than any integer under 72.[ 8] 48 , and contains the firstsix highly totient numbers1 ,2 ,4 ,8 , 12 and 24 as a subset of itsproper divisors .144 , or twice 72, is also highly totient, as is576 , thesquare of 24.[ 8] Euler's totient function φ(x ) over the first15 integers is 72.[ 9] perfect Harshad number indecimal (twenty-eighth), as it is divisible by the sum of its digits (9 ).[ 10]
72 is the second multiple of 12, after 48, that is not a sum oftwin primes .primes (13 + 17 + 19 + 23) ,[ 11] (5 + 7 + 11 + 13 + 17 + 19) .[ 12] 72 is the first number that can be expressed as the difference of the squares of primes in just two distinct ways:112 − 72 = 192 − 172 .[ 13] 72 is the sum of the first twosphenic numbers (30 ,42 ),[ 14] 12 , that is also theirabundance .[ 15] [ 16] 72 is themagic constant of the first non-normal, fullprime reciprocal magic square indecimal , based on1 / 17 [ 17] [ 18] 72 is the sum between60 and12 , the former being the secondunitary perfect number before6 (and the latter the smallest of only twosublime numbers ).[ 19] 72 is the number of distinct{7/2} magic heptagrams , all with a magic constant of 30.[ 20] 72 is the sum of the eighth row ofLozanić's triangle , and equal to the sum of the previous four rows (36, 20, 10, 6).[ 21] k 72 is the number ofdegrees in thecentral angle of aregular pentagon , which isconstructible with a compass and straight-edge. 72 plays a role in theRule of 72 ineconomics when approximating annualcompounding ofinterest rates of a round 6% to 10%, due in part to its high number of divisors.
InsideE n {\displaystyle \mathrm {E} _{n}} :
72 is the number ofvertices of thesix-dimensional 122 polytope , which also contains asfacets 720 edges ,702 polychoral 4-faces, of which270 arefour-dimensional 16-cells , and two sets of27 demipenteract 5 -faces. These 72 vertices are theroot vectors simple Lie group E 6 {\displaystyle \mathrm {E} _{6}} honeycomb under222 E 6 {\displaystyle \mathrm {E} _{6}} 122 is part of a family ofk22 3-3 duoprism , ofsymmetry order 72 and made of sixtriangular prisms . On the other hand,321 k21 semiregular polytope in theseventh dimension , also featuring a total of 7026 -faces of which 576 are6-simplexes and 126 are6-orthoplexes that contain60 edges and12 vertices, or collectively72 one-dimensional and two-dimensionalelements ; with126 the number ofroot vectors inE 7 {\displaystyle \mathrm {E} _{7}} 231 ∈k31 576 or 242 6-simplexes like321 . The triangular prism is the root polytope in thek21 family of polytopes, which is the simplest semiregular polytope, withk31 rooted in the analogous four-dimensionaltetrahedral prism that has four triangular prisms alongside two tetrahedra ascells . Thecomplex Hessian polyhedron inC 3 {\displaystyle \mathbb {C} ^{3}} complex triangular edges , as well as 27polygonal Möbius–Kantor faces and 27 vertices. It is notable for being thevertex figure of thecomplex Witting polytope , which shares240 vertices with the eight-dimensionalsemiregular 421 root vectors of thesimple Lie group E 8 {\displaystyle \mathrm {E} _{8}} There are 72compact andparacompact Coxeter groups of ranks four through ten: 14 of these are compact finite representations in onlythree-dimensional andfour-dimensional spaces, with the remaining 58 paracompact or noncompactinfinite representations in dimensions three through nine. These terminate with three paracompact groups in theninth dimension , of which the most important isT ~ 9 {\displaystyle {\tilde {T}}_{9}} semiregular hyperbolic honeycomb621 regular facets and the521 Euclidean honeycomb as itsvertex figure , which is the geometric representation of theE 8 {\displaystyle \mathrm {E} _{8}} T ~ 9 {\displaystyle {\tilde {T}}_{9}} over-extended formE 8 {\displaystyle \mathrm {E} _{8}} ++ tenth-dimensional symmetries of Lie algebraE 10 {\displaystyle \mathrm {E} _{10}}
72 lies between the8 th pair oftwin primes (71 ,73 ), where 71 is the largestsupersingular prime that is a factor of the largestsporadic group (thefriendly giant F 1 {\displaystyle \mathbb {F_{1}} } indexed member of adefinite quadratic integer matrix representative of allprime numbers [ 23] [ a] multiplicity ) inside all 194conjugacy classes ofF 1 {\displaystyle \mathbb {F_{1}} } [ 24] finite simple groups , whereE 6 {\displaystyle \mathrm {E} _{6}} E 7 {\displaystyle \mathrm {E} _{7}} E 8 {\displaystyle \mathrm {E} _{8}} exceptional groups that are part of sixteenfinite Lie groups that are alsosimple , or non-trivial groups whose onlynormal subgroups are thetrivial group and the groups themselves.[ b]
Seventy-two is also:
^ Where 71 is also the largest prime number less than 73 that is not a member of this set. ^ The only other finite simple groups are the infinite families ofcyclic groups andalternating groups . An exception is theTits group T {\displaystyle \mathbb {T} } non-strict group of Lie type that can otherwise more loosely classify as a 27th sporadic group. ^ Sloane, N. J. A. (ed.)."Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-06-15 .^ Sloane, N. J. A. (ed.)."Sequence A052486 (Achilles numbers - powerful but imperfect.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-10-22 .^ Sloane, N. J. A. (ed.)."Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-10-22 .^ Sloane, N. J. A. (ed.)."Sequence A081357 (Sublime numbers, numbers for which the number of divisors and the sum of the divisors are both perfect.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-06-15 .^ Sloane, N. J. A. (ed.)."Sequence A033950 (Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-06-15 .The sequence of refactorable numbers goes: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, ... ^ Sloane, N. J. A. (ed.)."Sequence A067128 (Ramanujan's largely composite numbers)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.^ Sloane, N. J. A. (ed.)."Sequence A000010 (Euler totient function.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-10-22 .^a b Sloane, N. J. A. (ed.)."Sequence A097942 (Highly totient numbers.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-10-22 .^ Sloane, N. J. A. (ed.)."Sequence A002088 (Sum of totient function.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-10-22 .^ Sloane, N. J. A. (ed.)."Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-10-22 .^ Sloane, N. J. A. (ed.)."Sequence A034963 (Sums of four consecutive primes.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-12-02 .^ Sloane, N. J. A. (ed.)."Sequence A127333 (Numbers that are the sum of 6 consecutive primes.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-12-02 .^ Sloane, N. J. A. (ed.)."Sequence A090788 (Numbers that can be expressed as the difference of the squares of primes in just two distinct ways.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-01-03 .^ Sloane, N. J. A. (ed.)."Sequence A007304 (Sphenic numbers: products of 3 distinct primes.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-02-13 .^ Sloane, N. J. A. (ed.)."Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-02-13 .^ Sloane, N. J. A. (ed.)."Sequence A033880 (Abundance of n, or (sum of divisors of n) - 2n.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-02-13 .^ Subramani, K. (2020)."On two interesting properties of primes, p, with reciprocals in base 10 having maximum period p - 1" (PDF) .J. Of Math. Sci. & Comp. Math .1 (2). Auburn, WA: S.M.A.R.T.:198– 200.doi :10.15864/jmscm.1204 (inactive 1 July 2025).eISSN 2644-3368 .S2CID 235037714 . {{cite journal }}: CS1 maint: DOI inactive as of July 2025 (link )^ Sloane, N. J. A. (ed.)."Sequence A007450 (Decimal expansion of 1/17.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2023-11-24 .^ Sloane, N. J. A. (ed.)."Sequence A005179 (Smallest number with exactly n divisors.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2024-03-11 .^ Sloane, N. J. A. (ed.)."Sequence A200720 (Number of distinct normal magic stars of type {n/2}.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-12-09 .^ Sloane, N. J. A. (ed.)."Sequence A005418 (...row sums of Losanitsch's triangle.)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved2022-10-22 .^ David Wells: The Penguin Dictionary of Curious and Interesting Numbers ^ Sloane, N. J. A. (ed.)."Sequence A154363 (Numbers from Bhargava's prime-universality criterion theorem)" .TheOn-Line Encyclopedia of Integer Sequences . OEIS Foundation.{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67,73 } ^ He, Yang-Hui ;McKay, John (2015). "Sporadic and Exceptional". p. 20.arXiv :1505.06742 math.AG ].^ Jami`at-Tirmidhi."The Book on Virtues of Jihad, Vol. 3, Book 20, Hadith 1663" .Sunnah.com - Sayings and Teachings of Prophet Muhammad (صلى الله عليه و سلم) . Retrieved2024-04-02 . ^ Kruglanski, Arie W.; Chen, Xiaoyan; Dechesne, Mark; Fishman, Shira; Orehek, Edward (2009)."Fully Committed: Suicide Bombers' Motivation and the Quest for Personal Significance" .Political Psychology .30 (3):331– 357.doi :10.1111/j.1467-9221.2009.00698.x .ISSN 0162-895X .JSTOR 25655398 . ^ W3C."CSS Units" .w3.org . RetrievedSeptember 28, 2024 . {{cite web }}: CS1 maint: numeric names: authors list (link )^ "Japan's 72 Microseasons" . 16 October 2015.
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