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A007970 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A007970 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A007970" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA007970">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A007970 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A007970 </div> <div class=seqname> Rhombic numbers. </div> </div> <div class=scorerefs> 19 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>3, 7, 8, 11, 15, 19, 23, 24, 27, 31, 32, 35, 40, 43, 47, 48, 51, 59, 63, 67, 71, 75, 79, 80, 83, 87, 88, 91, 96, 99, 103, 104, 107, 115, 119, 120, 123, 127, 128, 131, 135, 136, 139, 143, 151, 152, 159, 160, 163, 167, 168, 171, 175, 176, 179</div> <div class=seqdatalinks> (<a href="/A007970/list">list</a>; <a href="/A007970/graph">graph</a>; <a href="/search?q=A007970+-id:A007970">refs</a>; <a href="/A007970/listen">listen</a>; <a href="/history?seq=A007970">history</a>; <a href="/search?q=id:A007970&fmt=text">text</a>; <a href="/A007970/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline><a href="/A191856" title="First factor in happy factorization of n-th rhombic number.">A191856</a>(n) = <a href="/A007966" title="First factor in happy factorization of n.">A007966</a>(a(n)); <a href="/A191857" title="Second factor in happy factorization of n-th rhombic number.">A191857</a>(n) = <a href="/A007967" title="Second factor in happy factorization of n.">A007967</a>(a(n)). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jun 18 2011</div> <div class=sectline>This sequence gives the values d of the Pell equation x^2 - d*y^2 = +1 that have positive fundamental solutions (x0, y0) with odd y0. This was first conjectured and is proved provided Conway's theorem in the link is assumed and the proof of the conjecture stated in <a href="/A007869" title="Number of complementary pairs of graphs on n nodes. Also number of unlabeled graphs with n nodes and an even number of edges.">A007869</a>, given there in a W. Lang link, is used. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Sep 19 2015</div> <div class=sectline>For a proof of Conway's theorem on the happy number factorization see the W. Lang link (together with the link given under <a href="/A007969" title="Rectangular numbers.">A007969</a>). - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 04 2015</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Reinhard Zumkeller, <a href="/A007970/b007970.txt">Table of n, a(n) for n = 1..99</a></div> <div class=sectline>J. H. Conway, <a href="http://www.cs.uwaterloo.ca/journals/JIS/happy.html">On Happy Factorizations</a>, J. Integer Sequences, Vol. 1, 1998, #1.</div> <div class=sectline>Wolfdieter Lang, <a href="/A007970/a007970_1.pdf">Proof of a Theorem Related to the Happy Number Factorization.</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A191856" title="First factor in happy factorization of n-th rhombic number.">A191856</a>(n)*<a href="/A191857" title="Second factor in happy factorization of n-th rhombic number.">A191857</a>(n); <a href="/A007968" title="Type of happy factorization of n.">A007968</a>(a(n))=2. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jun 18 2011</div> <div class=sectline>a(n) is in the sequence if a(n) = D*E with positive integers D and E, such that E*U^2 - D*T^2 = 2 has an integer solution with U*T odd (without loss of generality one may take U and T positive). See the Conway link. D and E are given in <a href="/A191856" title="First factor in happy factorization of n-th rhombic number.">A191856</a> and <a href="/A191857" title="Second factor in happy factorization of n-th rhombic number.">A191857</a>, respectively. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 05 2015</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>r[b_, c_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + 2 == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, First[red], red]);</div> <div class=sectline>f[n_] := f[n] = If[! IntegerQ[Sqrt[n]], Catch[Do[{b, c} = bc; If[ (r0 = r[b, c]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]]; If[ (r0 = r[c, b]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]], {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]} ]]];</div> <div class=sectline><a href="/A007970" title="Rhombic numbers.">A007970</a> = Reap[ Table[ If[f[n] =!= Null, Print[f[n]]; Sow[f[n]]], {n, 1, 180}] ][[2, 1]](* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Jun 26 2012 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Haskell)</div> <div class=sectline>a007970 n = a007970_list !! (n-1)</div> <div class=sectline>a007970_list = filter ((== 2) . a007968) [0..]</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Oct 11 2015</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Every number belongs to exactly one of <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A007969" title="Rectangular numbers.">A007969</a>, <a href="/A007970" title="Rhombic numbers.">A007970</a>.</div> <div class=sectline>Cf. <a href="/A007968" title="Type of happy factorization of n.">A007968</a>.</div> <div class=sectline>Subsequence of <a href="/A000037" title="Numbers that are not squares (or, the nonsquares).">A000037</a>, <a href="/A002145" title="Primes of the form 4*k + 3.">A002145</a> is a subsequence.</div> <div class=sectline><a href="/A263008" title="First member T0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).">A263008</a> (T numbers), <a href="/A263009" title="Second member U0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).">A263009</a> (U numbers).</div> <div class=sectline>Sequence in context: <a href="/A047528" title="Numbers that are congruent to {0, 3, 7} mod 8.">A047528</a> <a href="/A069122" title="Numbers k such that the squarefree part of k is greater than the squarefree part of (k+1).">A069122</a> <a href="/A278519" title="a(n) = largest k for which A260731(k) = n.">A278519</a> * <a href="/A255342" title="Numbers n such that there are exactly two 1's in their factorial base representation (A007623).">A255342</a> <a href="/A332572" title="Numbers that are norm-deficient in Gaussian integers.">A332572</a> <a href="/A134258" title="Positions of 8 after decimal point in decimal expansion of 1/Pi.">A134258</a></div> <div class=sectline>Adjacent sequences: <a href="/A007967" title="Second factor in happy factorization of n.">A007967</a> <a href="/A007968" title="Type of happy factorization of n.">A007968</a> <a href="/A007969" title="Rectangular numbers.">A007969</a> * <a href="/A007971" title="INVERTi transform of central trinomial coefficients (A002426).">A007971</a> <a href="/A007972" title="Number of permutations that are 2 "block reversals" away from 12...n.">A007972</a> <a href="/A007973" title="Number of permutations that are n-2 "block reversals" away from 12...n.">A007973</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:J._H._Conway">J. H. Conway</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>159 and 175 inserted by <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Jun 26 2012</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified November 24 04:03 EST 2024. Contains 378053 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>