A007969 - OEIS

%I #63 Nov 09 2015 06:58:52

%S 2,5,6,10,12,13,14,17,18,20,21,22,26,28,29,30,33,34,37,38,39,41,42,44,

%T 45,46,50,52,53,54,55,56,57,58,60,61,62,65,66,68,69,70,72,73,74,76,77,

%U 78,82,84,85,86,89,90,92,93,94,95,97,98,101,102,105,106,108,109

%N Rectangular numbers.

%C A191854(n) = A007966(a(n)); A191855(n) = A007967(a(n)). - _Reinhard Zumkeller_, Jun 18 2011

%C It seems that D(n) = 4*a(n) gives precisely those even discriminants D from 4*A000037 of indefinite binary quadratic forms that have only improper solutions of the Pell equation x^2 - D*y^2 = +4. Conjecture tested for n = 1..66. Alternatively, the conjecture is that this sequence gives the r values for the Pell equation X^2 + r Y^2 = +1 whenever Y is even. See A261249 and A261250. - _Wolfdieter Lang_, Sep 16 2015

%C The proof of these two versions of the conjecture is given in the W. Lang link. - _Wolfdieter Lang_, Sep 19 2015 (revised Oct 03 2015)

%H Reinhard Zumkeller, <a href="/A007969/b007969.txt">Table of n, a(n) for n = 1..200</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="/A007969/a007969_2.pdf">Proof of a Conjecture Related to the 1-Happy Numbers.</a>

%F a(n) = A191854(n)*A191855(n); A007968(a(n)) = 1. - _Reinhard Zumkeller_, Jun 18 2011

%F a(n) is in the sequence if a(n) = C*B with integers B >= 1 and C >= 2, such that C*S^2 - B*R^2 = 1 has an integer solution (without loss of generality one may take S and R positive). See the Conway link. - _Wolfdieter Lang_, Sep 18 2015

%e From _Wolfdieter Lang_, Sep 18 2015: (Start)

%e a(1) = 5 = 5*1 and 5*1^2 - 1*2^2  = 1.

%e a(7) = 14 = 2*7 and 2*2^2 - 7*1^2 = 1. (End)

%t r[b_, c_] := (red = Reduce[x>0 && y>0 && b*x^2 + 1 == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, First[red], red]); f[128] = {}(* to speed up *); f[n_] := f[n] = If[IntegerQ[Sqrt[n]], {}, Do[c = n/b; If[(r0 = r[b, c]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; Return[{b, c, x0, y0}]], {b, Divisors[n] // Most}]]; A007969 = Reap[Table[Print[n, " ", f[n]]; If[f[n] != {} && f[n] =!= Null, Sow[n]], {n, 1, 130}]][[2, 1]] (* _Jean-François Alcover_, Jun 26 2012, updated Sep 18 2015 *)

%o (Haskell)

%o a007969 n = a007969_list !! (n-1)

%o a007969_list = filter ((== 1) . a007968) [0..]

%o -- _Reinhard Zumkeller_, Oct 11 2015

%Y Every number belongs to exactly one of A000290, A007969, A007970.

%Y Cf. A191854 (B numbers), A191855 (C numbers).

%Y Cf. A007966, A007967, A007968, A261249, A261250.

%Y Subsequence of A000037, A002144 is a subsequence.

%Y A263006 (R numbers), A263007 (S numbers).

%K nonn

%O 1,1

%A _J. H. Conway_