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You could also send an email to the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards" class="extiw" title="m:Special:MyLanguage/Stewards">stewards</a> <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/VRT" class="extiw" title="m:Special:MyLanguage/VRT">VRT</a> queue at <kbd>stewards@wikimedia.org</kbd> including all above details. </p> </div> <p>Other useful links: <a href="https://meta.wikimedia.org/wiki/Global_blocks" class="extiw" title="m:Global blocks">Global blocks</a> · <a href="/wiki/Help:I_have_been_blocked" title="Help:I have been blocked">Help:I have been blocked</a> </p> </div></li></ul><hr /> <div id="viewsourcetext">You can view and copy the source of this page:</div><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="en" dir="ltr" name="wpTextbox1">===Number theory=== 168 is the fourth [[Dedekind number]],<ref>{{Cite OEIS |A000372 |Dedekind numbers: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families. |access-date=2024-04-01 }}</ref> and one of sixty-five [[idoneal number]]s.<ref name="IdonealN">{{Cite web|url=https://oeis.org/A000926|title=Sloane's A000926 : Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers)|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2016-05-28}}</ref> It is one less than a [[Square number|square]] (13<sup>2</sup>), equal to the product of the first two [[perfect number]]s<ref>{{Cite OEIS |A000396 |Perfect numbers k: k is equal to the sum of the proper divisors of k. |access-date=2024-04-01 }}</ref> :<math>168 = 6 \times 28.</math> There are 168 [[prime number|primes]] less than [[1000 (number)|1000]].{{efn|1={{math|1=(168, 1000) }} un-inclusive corresponds to a [[Interval (mathematics)|range]] of 831 [[integer]]s, which is a value in equivalence with the composite index of 1000 = 10<sup>3</sup>.<ref name="CompNum" /> }} ==== Composite index ==== The 128th [[composite number]] is 168,<ref name="CompNum">{{Cite OEIS |A002808 |The composite numbers: numbers n of the form x*y for x greater than 1 and y greater than 1. |access-date=2024-04-05 }}</ref> one of a few numbers <math>n</math> in the list of composites whose [[Sequence|indices]] are the product of strings of [[Numerical digit|digits]] of <math>n</math> in [[decimal]] representation. The first nine <math>n</math> with this property are the following:<ref name="CompNum" />{{Bullet list |'''48''' as the '''32'''nd {{math|1=([[4]] × '''8''' = [[32 (number)|32]]) }}{{hair-space}}{{efn|1=32 is the twentieth composite. }} |[[68 (number)|68]] as the 48th {{math|1=([[6]] × '''8''' = [[48 (number)|48]]) }} |[[78 (number)|78]] is the 56th {{math|1=([[7]] × '''8''' = [[56 (number)|56]]) }} |[[88 (number)|88]] as the 64th {{math|1=('''8'''<sup>2</sup> = [[64 (number)|64]]) }} |[[98 (number)|98]] as the 72nd {{math|1=([[9]] × '''8''' = [[72 (number)|72]]) }} |[[128 (number)|128]] as the 96th {{math|1=([[12 (number)|12]] × '''8''' = [[96 (number)|96]]) }} |[[138 (number)|138]] as the 104th {{math|1=([[13 (number)|13]] × '''8''' = [[104 (number)|104]]) }} |[[158 (number)|158]] as the 120th {{math|1=([[15 (number)|15]] × '''8''' = [[120 (number)|120]]) }} |'''168''' as the '''128'''th {{math|1=([[16 (number)|16]] × '''8''' = [[128 (number)|128]]) }}{{hair-space}}{{efn|1=128 = 64 × 2 = 32 × 4, with 96 = 48 × 2, where also 168<sub>10</sub> = 120<sub>12</sub> (in [[duodecimal]]).<br />On the other hand, '''[[28 (number)|28]]''' is the 18th composite number,<ref name="CompNum" /> }} }} The next such number is 198 where {{math|1=[[19 (number)|19]] × '''8''' = [[152 (number)|152]] }}. The [[median]] between twenty-one integers {{math|1=[48, 68] }} is [[58 (number)|58]], where [[148 (number)|148]] is the median of forty-one integers {{math|1=[168, 128] }}. ==== Totient and sigma values ==== For the [[Euler totient]] <math>\varphi (n)</math> there is <math>\varphi (168) = 48</math>,<ref name="EulerToti">{{Cite OEIS |A000010 |Euler totient function phi(n): count numbers less than or equal to n and prime to n. |access-date=2024-04-05 }}</ref> where <math>\varphi (48) = 16</math> is also equivalent to the [[Divisor function|number of divisors]] of 168;<ref>{{Cite OEIS |A000005 |d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. |access-date=2024-04-05 }}</ref> only eleven numbers have a totient of 48:{{math|1={'''[[65 (number)|65]]''', 104, 105, 112, 130, 140, 144, 156, '''168''', 180, '''210'''} }}.<ref name="EulerToti" />{{efn|1=The latter (210) is the 20th [[triangle number]].<ref>{{Cite OEIS |A000217 |Triangular numbers: 0 + 1 + 2 + ... + n. |access-date=2024-04-09 }}</ref> }} The [[Divisor function|number of divisors]] of 168 is 16,<ref>{{Cite OEIS |A000005 |d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. |access-date=2024-04-05 }}</ref> making it a [[largely composite number]].<ref name="OEIS-A067128">{{Cite OEIS|A067128|Ramanujan's largely composite numbers}}</ref> [[400 (number)#400 to 409|408]],{{efn|1=[[505 (number)|505]], which is the [[magic constant]] of a <math>10 \times 10</math> [[magic square]],<ref>{{Cite OEIS |A006003 |a(n) as n*(n^2 + 1)/2. |access-date=2024-04-08 }}</ref> is the 408th composite number. }} with a different permutation of the digits {{math|1={[[0]], [[4]], [[8]]} }} where [[48 (number)|048]] is 48, has an [[Euler totient|totient]] of 128. So does the [[Divisor function|sum-of-divisors]] of 168,<ref name="A000203" /> :<math>\sigma (168) = 480 = 2^{5} \times 3 \times 5</math> as one of nine numbers total to have a totient of 128.<ref name="EulerToti" /> 48 sets the sixteenth record for sum-of-divisors of positive integers (of [[124 (number)|124]]), and the seventeenth record value is 168,<ref>{{Cite OEIS |A034885 |Record values of sigma(n). |access-date=2024-07-06 }}</ref> from six numbers ([[60 (number)|60]], [[78 (number)|78]], [[92 (number)|92]], [[123 (number)|123]], [[143 (number)|143]], and [[167 (number)|167]]).<ref name="A000203" /> The difference between 168 and 48 is the [[factorial]] of five ([[120 (number)|120]]), where their sum is the [[Cubic number|cube]] of six ([[216 (number)|216]]). ==== Idoneal number ==== [[Leonhard Euler]] noted 65 [[idoneal number]]s (the most known, of only a maximum possible of two more), such that <math>x^{2} +Dy^{2}</math> for an integer <math>D</math>, expressible in only one way, yields a [[prime power]] or twice a prime power.<ref name="IdonealN" /><ref>{{Cite journal |last=Euler |first=Leonard |title=Illustratio paradoxi circa progressionem numerorum idoneorum sive congruorum |journal=Nova Acta Academiae Scientarum Imperialis Petropolitinae |publisher=[[Russian Academy of Sciences]] |volume=15 |year=1806 |pages=29–32 |arxiv=math/0507352 |s2cid=118287274 }}</ref> Of these, 168 is the forty-fourth, where the smallest number to not be idoneal is the fifth prime number [[11 (number)|11]].<ref name="IdonealN" /> The largest such number 1848 (that is equivalent with the number of [[Edge (geometry)|edges]] in the [[Union (set theory)|union]] of two [[cycle graph]]s of [[Group order|order]] [[42 (number)|42]])<ref>{{Cite OEIS |A005563 |a(n) as n*(n+2) equal to (n+1)^2 - 1. |access-date=2024-04-09 }}</ref> contains a total of thirty-two divisors whose arithmetic mean is '''180'''<ref name="ArithNumO">{{Cite OEIS |A003601 |Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n). |access-date=2023-07-16 }}</ref><ref name="ArithNum1">{{Cite OEIS |A102187 |Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer). |access-date=2023-07-16 }}</ref> (the second-largest number to have a totient of 48).<ref name="EulerToti" /> Preceding 1848 in the list of idoneal numbers is [[1365 (number)|1365]],{{efn|1={{math|1=1365 ÷ 3 = [[400 (number)#450s|455]] }} is the sum of (the first) ten terms in the sequence of numbers {{math|1=''k'' }} ∈ {{math|1= {1, 2, 3, 4, 7, 8, 16, 31, 127, 256} }} such that {{math|1=''k'' }} and {{math|1=''k'' + 1 }} are prime powers.<ref>{{Cite OEIS |A006549 |Numbers k such that k and k+1 are prime powers. |access-date=2024-04-09 }}</ref> }} whose arithmetic mean of [[divisor]]s is equal to 168<ref name="ArithNumO" /><ref name="ArithNum1" /> (while 1365 has a totient of [[576 (number)|576]] = 24<sup>2</sup>). Where 48 is the 27th ideoneal number, 408 is the 58th.<ref name="IdonealN" />{{efn|1= 840, with thirty-two divisors (the number with the largest number of divisors less than 1000), is the fourth-largest idoneal number. 88, 78, 58, 28, and 18 are also idoneal numbers, including 210 and 105 (numbers with totients of 48).<ref name="IdonealN" /> }} On the other hand, the total count of known idoneal numbers (65), that is also equal to the sum of ten integers {{math|1=[2, ..., 11] }}, has a sum-of-divisors of [[84 (number)|84]] (or, one-half of 168).<ref name="A000203">{{Cite OEIS |A000203 |The sum of divisors of n. |access-date=2024-04-09 }}</ref> ==== Numbers of the form 2<sup>''n''</sup> ==== In [[Decimal|base 10]], 168 is the largest of ninety-two known <math>n</math> such that <math>2^{n}</math> does not contain all [[numerical digit]]s from that base (i.e. [[0]], [[1]], [[2]], ..., [[9]]).<ref name="2nb10">{{Cite web|url=https://oeis.org/A130696|title=Sloane's A130696: Numbers k such that 2^k does not contain all ten decimal digits.|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2022-12-19}}</ref> <math>2^{68}</math> is the first number to have such an expression where between the next two <math>n</math> is an [[Interval (mathematics)|interval]] of ten integers: {{math|1=<nowiki>[</nowiki>[[70 (number)|70]], [[79 (number)|79]]<nowiki>]</nowiki> }};<ref name="2nb10" /> the median values between these are (75, [[74 (number)|74]]), where the smaller of these two values represents the composite index of [[100]].<ref name="CompNum" />{{efn|1=In the iterative list of the {{math|1=''A''(''n'') }}-th composite number with {{math|1=''A''(1) = 11 }} where {{math|1=''A''(''n + 1'') = ''A''(''n'') }}, the first few elements are <br />{{math|1=11, 20, 32, 48, 68, 93, 124, ... }}<ref>{{Cite OEIS |A059407 |a(n+1) as the a(n)-th composite number, with a(1) equal to 11. |access-date=2024-04-08 }}</ref><br />which is preceded at 11 with the analogous list of successive [[super-prime]]s<ref>{{Cite OEIS |A006450 |Prime-indexed primes: primes with prime subscripts. |access-date=2024-04-08 }}</ref> and primes<ref>{{Cite OEIS |A000040 |The prime numbers. |access-date=2024-04-08 }}</ref> {{math|1=11, 5, 3, 2, 1 }} (if the [[1|unit]] is a zeroth prime).<br /> The sum of these elements {{math|1=1, 2, 3, 5, 11, 20, 32}} is 74, with {{math|1=32 + 68 = 100 }}, and {{math|1=48 }} in between. }} ==== Cunningham number ==== As a number of the form <math>b^n\pm1</math> for positive integers <math>n</math>, <math>b</math> and <math>b</math> not a [[perfect power]], 168 is the thirty-second [[Cunningham number]],<ref>{{Cite OEIS |A080262 |Cunningham numbers: of the form a^b +- 1, where a, b are greater than or equal to 2. |access-date=2024-04-20 }}</ref> where it is one less than a [[Square number|square]]: :<math>168 = 13^{2} - 1.</math> On the other hand, 168 is one more than the third member of the fourth [[Cunningham chain|chain of nearly doubled primes]] of the first kind {{math|1={[[41 (number)|41]], [[83 (number)|83]], '''167'''} }},<ref>{{Cite OEIS |A005602 |Smallest prime beginning a complete Cunningham chain of length n (of the first kind). |access-date=2024-04-20 }}</ref><ref>{{Cite OEIS |A348855 |a(1) is 1. If a(n) is prime, a(n+1) is 2*a(n) + 1. If a(n) is not prime, a(n+1) is the least prime not already in the sequence. |access-date=2024-04-20 }}</ref> where [[167 (number)|167]] represents the thirty-ninth [[Prime number|prime]]<ref>{{Cite OEIS |A000040 |The prime numbers. |access-date=2024-04-20 }}</ref> (with [[39 (number)|39]] × [[2]] = 78). The smallest such chain is {{math|1={2, 5, '''11''', 23, [[47 (number)|47]]} }}. ==== Eisenstein series ==== 168 is also coefficient four in the expansion of [[Eisenstein series]] <math>E_{2}</math>,<ref>{{Cite OEIS |A006352 |Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2). |access-date=2024-04-02 }}</ref> which also includes 144 and 96 (or 48 × 2) as the fifth and third coefficients, respectively — these have a sum of [[240 (number)|240]], which follows 144 and [[187 (number)|187]] in the list of successive composites <math>A(n + 1) = A(n)</math>;<sup>cf.</sup><ref name="CompNum"/> the latter holds a sum-of-divisors of [[216 (number)|216]] = [[6]]<sup>[[3]]</sup>,<ref name="A000203" /> which is the 168th composite number.<ref name="CompNum"/> </textarea><div class="templatesUsed"></div><p id="mw-returnto">Return to <a href="/wiki/168_(number)" title="168 (number)">168 (number)</a>.</p> <noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/wiki/168_(number)">https://en.wikipedia.org/wiki/168_(number)</a>"</div></div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=168_(number)&action=edit&section=2&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-zjw24","wgBackendResponseTime":263,"wgPageParseReport":{"limitreport":{"cputime":"0.057","walltime":"0.079","ppvisitednodes":{"value":418,"limit":1000000},"postexpandincludesize":{"value":17740,"limit":2097152},"templateargumentsize":{"value":6556,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":469,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 57.023 1 -total"," 99.83% 56.928 2 Template:Blocked_text"," 41.50% 23.666 1 Template:Colocationwebhost"," 39.32% 22.419 2 Template:Replace"," 37.70% 21.495 1 Template:Hidden"," 16.14% 9.206 1 Template:Tlx"," 14.48% 8.256 1 Template:Hidden_begin"," 3.37% 1.920 1 Template:Hidden_end"," 2.39% 1.364 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake"," 1.77% 1.009 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake-email-steward"]},"scribunto":{"limitreport-timeusage":{"value":"0.013","limit":"10.000"},"limitreport-memusage":{"value":1043305,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-5c59558b9d-zjw24","timestamp":"20241201111615","ttl":2592000,"transientcontent":false}}});});</script> </body> </html>