编辑“︁69”︁（章节）

== 在数学中 ==
69是[[半素数]]，因为它是两个[[素数]]（[[3]]和[[23]]）的乘积，也是[[67]]和[[71]]这兩個連續奇[[質數]]的[[算術平均數]]<ref>{{cite web|url=https://oeis.org/A001358|title=A001358: Semiprimes (or biprimes): products of two primes.|last1=Neil|first1=Sloane|last2=Guy|first2=R. K.|author2-link=Richard K. Guy|date=2010-08-22|website=On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2019-04-19|archive-url=https://web.archive.org/web/20190419095507/https://oeis.org/A001358|url-status=live}}</ref><ref>{{cite web|url=https://oeis.org/A024675|first=Clark|last=Kimberling|author-link=Clark Kimberling|date=n.d.|title=A024675: Average of two consecutive odd primes.|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2024-03-04|archive-url=https://web.archive.org/web/20240304232723/https://oeis.org/A024675|url-status=live}}</ref>。69不能被1以外的任何[[平方数]]整除，因此它是无平方因子数<ref>{{cite web|url=https://oeis.org/A005117|first=Neil|last=Sloane|date=n.d.|title=A005117: Squarefree numbers: numbers that are not divisible by a square greater than 1.|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2019-05-27|archive-url=https://web.archive.org/web/20190527172638/http://oeis.org/A005117|url-status=live}}</ref>。69亦是[[布卢姆数]]，因为69的两个因数都是高斯素数，并且位在[[烏拉姆數列|乌拉姆数列]]，是数列中先前出现的两个不同的乌拉姆数的和{{efn|1=根据乌拉姆序列的定义，3是乌拉姆数（1 + 2），4是乌拉姆数（1 + 3）。5 不是乌拉姆数，因为 5 = 1 + 4 = 2 + 3。69是乌拉姆数，因为它是16 + 53的和；16和53都是乌拉姆数<ref name=Gupta>{{cite book|first=Shyam Sunder|last=Gupta|editor-last=Wenpeng|editor-first=Zhang|year=2009|chapter=Smarandache sequence of Ulam numbers|title=Research on Number Theory and Smarandache Notions: Proceedings of the Fifth International Conference on Number Theory and Smarandache Notions|url=https://archive.org/details/multimedia-larga-9781599730882|publisher=Hexis|isbn=9781599730882|page=[https://archive.org/details/multimedia-larga-9781599730882/page/n86 78]}}</ref> <ref>{{cite journal|last=Recaman|first=Bernardo|year=1973|title=Questions on a sequence of Ulam|url=https://archive.org/details/sim_american-mathematical-monthly_1973-10_80_8/page/918|journal=[[American Mathematical Monthly]]|publisher=[[Mathematical Association of America]]|issue=8|volume=80|pages=919–920|doi=10.2307/2319404|jstor=2319404 }}</ref>。}}<ref name=Gupta>{{cite book|first=Shyam Sunder|last=Gupta|editor-last=Wenpeng|editor-first=Zhang|year=2009|chapter=Smarandache sequence of Ulam numbers|title=Research on Number Theory and Smarandache Notions: Proceedings of the Fifth International Conference on Number Theory and Smarandache Notions|url=https://archive.org/details/multimedia-larga-9781599730882|publisher=Hexis|isbn=9781599730882|page=[https://archive.org/details/multimedia-larga-9781599730882/page/n86 78]}}</ref><ref>{{cite web|url=https://oeis.org/A016105|first=Robert G.|last=Wilson|date=n.d.|title=A016105: Blum integers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2018-10-17|archive-url=https://web.archive.org/web/20181017021517/http://oeis.org/A016105|url-status=live}}</ref>。69是[[亏数]]，因为它的[[真因數和|真约数和]]（不包括自身的）小于自身<ref>{{cite web|url=https://oeis.org/A005100|first1=Neil|last1=Sloane|first2=Stefan|last2=Steinerberger|date=2006-03-31|title=A005100: Deficient numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2023-03-22|archive-url=https://web.archive.org/web/20230322182824/https://oeis.org/A005100|url-status=live}}</ref>。作为一个[[因数|正因数]]的[[算术平均值]]也是整数的整数，69亦为算术数<ref>{{cite web|url=https://oeis.org/A003601|last1=Sloane|first1=Neil|last2=Bernstein|first2=Mira|date=2006-04-03|title=A003601: Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2024-03-26|archive-url=https://web.archive.org/web/20240326083848/https://oeis.org/A003601|url-status=live}}</ref>。69是[[有理直角三角面積數|有理直角三角面积数]]（一个正整数，其面积等于三条有理数边的直角三角形），亦为可服从数<ref>{{cite journal|first1=Ronald|last1=Alter|first2=Thaddeus B.|last2=Curtz|date=January 1974|title=A Note on Congruent Numbers|url=https://archive.org/details/sim_mathematics-of-computation_1974-01_28_125/page/304|journal=[[Mathematics of Computation]]|publisher=[[American Mathematical Society]]|volume=28|number=125|pages=304–305|doi=10.2307/2005838|jstor=2005838}}</ref><ref>{{cite web|url=https://oeis.org/A100832|last=Beedassy|first=Lekraj|date=2005-01-07|title=A100832: Amenable numbers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2023-12-19|archive-url=https://web.archive.org/web/20231219114919/https://oeis.org/A100832|url-status=live}}</ref>。69可以用多种方式表示为连续正整数的和，因此它是一个[[礼貌数]]<ref>{{cite web|url=https://oeis.org/A138591|first1=Vladimir Joseph Stephan|last1=Orlovsky|first2=Carl R.|last2=White|date=2009-07-22|title=A138591: Sums of two or more consecutive nonnegative integers|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2019-03-12|archive-url=https://web.archive.org/web/20190312031016/http://oeis.org/A138591|url-status=live}}</ref>。69是一个[[幸运数]]，因为它是从1开始，在自然数序列中重复移除每n个数字后剩下的自然数{{efn|其中n是列表中最后一个幸存数字之后的下一个数字；数字列表中（1到无穷大）的每个第二个数字（所有偶数）首先被消除（1、3、5、7、9、11……），每第三个数字（1、3、7、9……），然后每第七个数字被消除，依此类推<ref>{{cite book|last=Giblin|first=P[eter] J.|author-link=Peter Giblin|year=1993|title=Primes and Programming|url=https://archive.org/details/primesprogrammin0000gibl|publisher=[[Cambridge University Press]]|isbn=9780521409889|page=[https://archive.org/details/primesprogrammin0000gibl/page/n82 67]}}</ref>。}}<ref>{{cite web|url=https://oeis.org/A002808|title=A002808: Composite numbers|last=Neil|first=Sloane|date=2010-12-16|website=On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2024-05-11|archive-url=https://web.archive.org/web/20240511171822/https://oeis.org/A002808|url-status=live}}</ref><ref>{{cite web|url=https://oeis.org/A000959|title=A000959: Lucky numbers|last=Neil|first=Sloane|date=2008-03-07|website=On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2019-04-19|archive-url=https://web.archive.org/web/20190419095457/https://oeis.org/A000959|url-status=live}}</ref>。

在[[十进制]]中，69的[[平方]]（4761）和立方（328 509）使用0–9中的每个数字恰好一次（即唯一一个）<ref>{{cite book|last=Wells|first=David|year=1997|title=[[The Penguin Dictionary of Curious and Interesting Numbers]]|edition=2|publisher=[[Penguin Books]]|isbn=0-14-008029-5|page=[https://archive.org/details/penguindictionar0000well_f3y1/page/n121 100]}}</ref><ref>{{cite book|last=Barbeau|first=Edward|author-link=Edward Barbeau|year=1997|title=Power Play|url=https://archive.org/details/powerplay0000barb|publisher=Mathematical Association of America|isbn=9780883855232|page=[https://archive.org/details/powerplay0000barb/page/n139 126]}}</ref>。它也是阶乘小于1[[古戈尔]]的最大数。在许多手持式[[科学计算器]]和[[图形计算器]]上，69! (1.711224524{{e|98}}) 是由于内存限制而可以计算的最高阶乘<ref>{{cite book|first=David Alexander|last=Brannan|year=2006|title=A First Course in Mathematical Analysis|publisher=[[Cambridge University Press]]|isbn=9781139458955|page=303}}</ref>。在其[[二进制]]展开式1000101中<ref>{{cite book|last=Konheim|first=Alan G.|year=2007|title=Computer Security and Cryptography|publisher=Wiley|isbn=9780470083970|page=382}}</ref>，69等于八进制中的105，而105等于[[十六进制]]中的69（同一属性可应用于从[[64]]到69的间的所有数字）<ref>{{cite book|last=Topham|first=Douglas W.|year=2012|title=A System V Guide to UNIX and XENIX|publisher=[[Springer New York]]|isbn=9781461232469|page=78}}</ref><ref name="Holmay">{{cite book|last=Holmay|first=Patrick|year=1998|chapter=ASCII Character Set (Continued)|title=The OpenVMS User's Guide|url=https://archive.org/details/openvmsusersguid0000holm|publisher=[[Elsevier Science]]|isbn=9781555582036|page=[https://archive.org/details/openvmsusersguid0000holm/page/n291 272]}}</ref>。在[[计算 (计算机科学)|计算机科学计算]]中，69等于[[三进制]]（基数为3）中的2120；[[六进制]]（基数为6）中的153；以及[[十二进制]]（基数为12）中的59<ref>{{cite book|last1=Clifford|first1=Jerrold R.|last2=Clifford|first2=Martin|year=1974|title=Computer Mathematics Handbook|url=https://archive.org/details/computermathemat0000clif|publisher=[[Allyn & Bacon]]|page=[https://archive.org/details/computermathemat0000clif/page/n297 276]}}</ref><ref>{{cite book|last=Scott|first=Norman Ross|year=1960|title=Analog and Digital Computer Technology|url=https://archive.org/details/analogdigitalcom0000scot_n4s0|publisher=[[McGraw-Hill]]|page=[https://archive.org/details/analogdigitalcom0000scot_n4s0/page/n238 221]}}</ref><ref>{{cite book|last=Meyer|first=Jerome S.|year=1963|title=More Fun with Mathematics|publisher=[[Gramercy Publishing Company]]|page=73}}</ref>。

从视觉上讲，在[[阿拉伯数字]]中，69为一旋转对称数，因为无论正看还是反看，它看起来都相同<ref>{{cite book|first=Elena|last=Deza|year=2013|title=Perfect And Amicable Numbers|publisher=World Scientific|isbn=9789811259647|page=390}}</ref>。69为一中心四面体数，该数形似金字塔，底部为三角形，其他所有点都在底部上方层层排列，形成四面体形<ref>{{cite book|first1=Elena|last1=Deza|first2=Michel|last2=Deza|author1-link=Elena Deza|author2-link=Michel Deza|year=2012|title=Figurative Numbers|publisher=[[World Scientific]]|isbn=9789814355483|pages=126–127}}</ref>。69也是一个有害数，因为当它转写成二进制时，1的个数为质数；它也是一个[[可惡數|可恶数]]，因为它是一个正整数，在二进制展开式中，1的个数为奇数<ref>{{cite web|url=https://oeis.org/A052294|last=Gow|first=Jeremy|date=2000-02-08|title=A052294: Pernicious numbers: numbers with a prime number of 1's in their binary expansion|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22}}</ref><ref>{{cite web|url=https://oeis.org/A000069|first=Neil|last=Sloane|date=n.d.|title=A000069: Odious numbers: numbers with an odd number of 1's in their binary expansion|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2024-04-22|archive-date=2018-09-11|archive-url=https://web.archive.org/web/20180911114600/https://oeis.org/A000069|url-status=live}}</ref>。