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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A191854(n) = A007966(a(n)); A191855(n) = A007967(a(n)). - Reinhard Zumkeller, Jun 18 2011
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
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
FORMULA
a(n) = A191854(n)*A191855(n); A007968(a(n)) = 1. - Reinhard Zumkeller, Jun 18 2011
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
From Wolfdieter Lang, Sep 18 2015: (Start)
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]] (* Jean-François Alcover, Jun 26 2012, updated Sep 18 2015 *)
PROG
(Haskell)
a007969 n = a007969_list !! (n-1)
a007969_list = filter ((== 1) . a007968) [0..]
-- Reinhard Zumkeller, Oct 11 2015
CROSSREFS
Every number belongs to exactly one of A000290, A007969, A007970.
Cf. A191854 (B numbers), A191855 (C numbers).
Subsequence of A000037, A002144 is a subsequence.
A263006 (R numbers), A263007 (S numbers).
Sequence in context: A064572 A032399 A276496 * A187902 A187231 A162340
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified July 23 12:38 EDT 2024. Contains 374549 sequences. (Running on oeis4.)