This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007970 #42 Apr 14 2020 05:44:35
%S A007970 3,7,8,11,15,19,23,24,27,31,32,35,40,43,47,48,51,59,63,67,71,75,79,80,
%T A007970 83,87,88,91,96,99,103,104,107,115,119,120,123,127,128,131,135,136,
%U A007970 139,143,151,152,159,160,163,167,168,171,175,176,179
%N A007970 Rhombic numbers.
%C A007970 A191856(n) = A007966(a(n)); A191857(n) = A007967(a(n)). - _Reinhard Zumkeller_, Jun 18 2011
%C A007970 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 A007869, given there in a W. Lang link, is used. - _Wolfdieter Lang_, Sep 19 2015
%C A007970 For a proof of Conway's theorem on the happy number factorization see the W. Lang link (together with the link given under A007969). - _Wolfdieter Lang_, Oct 04 2015
%H A007970 Reinhard Zumkeller, <a href="/A007970/b007970.txt">Table of n, a(n) for n = 1..99</a>
%H A007970 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.
%H A007970 Wolfdieter Lang, <a href="/A007970/a007970_1.pdf">Proof of a Theorem Related to the Happy Number Factorization.</a>
%F A007970 a(n) = A191856(n)*A191857(n); A007968(a(n))=2. - _Reinhard Zumkeller_, Jun 18 2011
%F A007970 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 A191856 and A191857, respectively. - _Wolfdieter Lang_, Oct 05 2015
%t A007970 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]);
%t A007970 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]]} ]]];
%t A007970 A007970 = Reap[ Table[ If[f[n] =!= Null, Print[f[n]]; Sow[f[n]]], {n, 1, 180}] ][[2, 1]](* _Jean-François Alcover_, Jun 26 2012 *)
%o A007970 (Haskell)
%o A007970 a007970 n = a007970_list !! (n-1)
%o A007970 a007970_list = filter ((== 2) . a007968) [0..]
%o A007970 -- _Reinhard Zumkeller_, Oct 11 2015
%Y A007970 Every number belongs to exactly one of A000290, A007969, A007970.
%Y A007970 Cf. A007968.
%Y A007970 Subsequence of A000037, A002145 is a subsequence.
%Y A007970 A263008 (T numbers), A263009 (U numbers).
%K A007970 nonn
%O A007970 1,1
%A A007970 _J. H. Conway_
%E A007970 159 and 175 inserted by _Jean-François Alcover_, Jun 26 2012