A263006 -id:A263006 - cp's OEIS frontend

A007969 Rectangular numbers.

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

Offset: 1

J. H. Conway

nonn

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)

			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)

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 1-Happy Numbers.

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).

a007969 n = a007969_list !! (n-1)
a007969_list = filter ((== 1) . a007968) [0..]
-- Reinhard Zumkeller, Oct 11 2015

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 *)

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

A263007 Second member S0(n) of the smallest positive pair (R0(n), S0(n)) for the n-th 1-happy number couple (B(n), C(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 4, 13, 1, 2, 3, 1, 1, 1, 5, 1, 5, 3, 78, 1, 5, 25, 3, 3, 1, 2, 13, 2, 3805, 4, 1, 1, 1, 36, 3, 1, 125, 5, 85, 4, 3, 1, 1, 41, 11, 53, 1, 12, 14, 732, 2, 569, 5, 1, 1, 1, 389, 13, 851525, 1, 2, 2, 73, 3, 13, 5, 51

Offset: 1

Wolfdieter Lang, Oct 28 2015

nonn

See A263007. C(n)*a(n)^2 - B(n)*A263007(n)^2 = +1, n >= 1, with the 1-happy couple (B(n), C(n)) = (A191854(n), A191855(n)).

In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = S0(n) numbers appear for the t = 1 rows in column w.

			n = 4: 1-happy number A007969(4) = 10 = 1*10 = A191854(4)*A191855(4). 10*a(4)^2 - 1*A263006(4)^2 = 10*1^2 - 1*3^2 = +1. This is the smallest positive solution for given (B, C) = (1, 10).

J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.

Cf. A007969, A191854, A191855, A191860, A263006, A262025, A261250.

A191855(n)*a(n)^2 - A191854(n)*A263006(n)^2 = +1, and A263006(n) with a(n) is the smallest positive solution for the given 1-happy couple (A191854(n), A191855(n)).

cp's OEIS Frontend

A007969 Rectangular numbers.

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