

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

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..200
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Wolfdieter Lang, Proof of a Conjecture Related to the 1Happy Numbers.


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


PROG

(Haskell)
a007969 n = a007969_list !! (n1)
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).
Cf. A007966, A007967, A007968, A261249, A261250.
Subsequence of A000037, A002144 is a subsequence.
A263006 (R numbers), A263007 (S numbers).
Sequence in context: A064572 A032399 A276496 * A187902 A187231 A162340
Adjacent sequences: A007966 A007967 A007968 * A007970 A007971 A007972


KEYWORD

nonn


AUTHOR

J. H. Conway


STATUS

approved



