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%I #42 Apr 14 2020 05:44:35
%S 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 83,87,88,91,96,99,103,104,107,115,119,120,123,127,128,131,135,136,
%U 139,143,151,152,159,160,163,167,168,171,175,176,179
%N Rhombic numbers.
%C A191856(n) = A007966(a(n)); A191857(n) = A007967(a(n)). - _Reinhard Zumkeller_, Jun 18 2011
%C 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 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 Reinhard Zumkeller, <a href="/A007970/b007970.txt">Table of n, a(n) for n = 1..99</a>
%H 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 Wolfdieter Lang, <a href="/A007970/a007970_1.pdf">Proof of a Theorem Related to the Happy Number Factorization.</a>
%F a(n) = A191856(n)*A191857(n); A007968(a(n))=2. - _Reinhard Zumkeller_, Jun 18 2011
%F 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 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 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 = Reap[ Table[ If[f[n] =!= Null, Print[f[n]]; Sow[f[n]]], {n, 1, 180}] ][[2, 1]](* _Jean-François Alcover_, Jun 26 2012 *)
%o (Haskell)
%o a007970 n = a007970_list !! (n-1)
%o a007970_list = filter ((== 2) . a007968) [0..]
%o -- _Reinhard Zumkeller_, Oct 11 2015
%Y Every number belongs to exactly one of A000290, A007969, A007970.
%Y Cf. A007968.
%Y Subsequence of A000037, A002145 is a subsequence.
%Y A263008 (T numbers), A263009 (U numbers).
%K nonn
%O 1,1
%A _J. H. Conway_
%E 159 and 175 inserted by _Jean-François Alcover_, Jun 26 2012
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