A007970 Rhombic numbers.

3, 7, 8, 11, 15, 19, 23, 24, 27, 31, 32, 35, 40, 43, 47, 48, 51, 59, 63, 67, 71, 75, 79, 80, 83, 87, 88, 91, 96, 99, 103, 104, 107, 115, 119, 120, 123, 127, 128, 131, 135, 136, 139, 143, 151, 152, 159, 160, 163, 167, 168, 171, 175, 176, 179

Offset: 1

Author

J. H. Conway

Keywords

• nonn

Comments

A191856(n) = A007966(a(n)); A191857(n) = A007967(a(n)). - Reinhard Zumkeller, Jun 18 2011

This sequence gives the values d of the Pell equation x^2 - d*y^2 = +1 that have positive fundamental solutions (x0, y0) with odd y0. This was first conjectured and is proved provided Conway's theorem in the link is assumed and the proof of the conjecture stated in A007869, given there in a W. Lang link, is used. - Wolfdieter Lang, Sep 19 2015

For a proof of Conway's theorem on the happy number factorization see the W. Lang link (together with the link given under A007969). - Wolfdieter Lang, Oct 04 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..99
• J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
• Wolfdieter Lang, Proof of a Theorem Related to the Happy Number Factorization.

Crossrefs

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

Cf. A007968.

Subsequence of A000037, A002145 is a subsequence.

A263008 (T numbers), A263009 (U numbers).

Programs

Haskell

a007970 n = a007970_list !! (n-1)
a007970_list = filter ((== 2) . a007968) [0..]
-- Reinhard Zumkeller, Oct 11 2015

Mathematica

r[b_, c_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + 2 == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, First[red], red]);
f[n_] := f[n] = If[! IntegerQ[Sqrt[n]], Catch[Do[{b, c} = bc; If[ (r0 = r[b, c]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]]; If[ (r0 = r[c, b]) =!= False, {x0, y0} = {x, y} /. ToRules[r0]; If[OddQ[x0] && OddQ[y0], Throw[n]]], {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]} ]]];
A007970 = Reap[ Table[ If[f[n] =!= Null, Print[f[n]]; Sow[f[n]]], {n, 1, 180}] ][[2, 1]](* Jean-François Alcover, Jun 26 2012 *)

Formula

a(n) = A191856(n)*A191857(n); A007968(a(n))=2. - Reinhard Zumkeller, Jun 18 2011

a(n) is in the sequence if a(n) = D*E with positive integers D and E, such that E*U^2 - D*T^2 = 2 has an integer solution with U*T odd (without loss of generality one may take U and T positive). See the Conway link. D and E are given in A191856 and A191857, respectively. - Wolfdieter Lang, Oct 05 2015

Extensions

159 and 175 inserted by Jean-François Alcover, Jun 26 2012