

A007969


Rectangular numbers.


21



2, 5, 6, 10, 12, 13, 14, 17, 18, 20, 21, 22, 26, 28, 29, 30, 33, 34, 37, 38, 39, 41, 42, 44, 45, 46, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 82, 84, 85, 86, 89, 90, 92, 93, 94, 95, 97, 98, 101, 102, 105, 106, 108, 109
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OFFSET

1,1


COMMENTS

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
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)


LINKS



FORMULA

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


EXAMPLE

a(1) = 5 = 5*1 and 5*1^2  1*2^2 = 1.
a(7) = 14 = 2*7 and 2*2^2  7*1^2 = 1. (End)


MATHEMATICA

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]] (* JeanFrançois Alcover, Jun 26 2012, updated Sep 18 2015 *)


PROG

(Haskell)
a007969 n = a007969_list !! (n1)
a007969_list = filter ((== 1) . a007968) [0..]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



