

A007970


Rhombic numbers.


12



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
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OFFSET

1,1


COMMENTS

Subsequence of A000037; A007968(a(n))=2; A002145 is a subsequence;
A191856(n) = A007966(a(n)); A191857(n) = A007967(a(n));
a(n) = A191856(n)*A191857(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

Table of n, a(n) for n=1..55.
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.


FORMULA

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


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]](* JeanFrançois Alcover, Jun 26 2012 *)


CROSSREFS

Every number belongs to exactly one of A000290, A007969, A007970.
Sequence in context: A078466 A047528 A069122 * A255342 A134258 A028972
Adjacent sequences: A007967 A007968 A007969 * A007971 A007972 A007973


KEYWORD

nonn,changed


AUTHOR

J. H. Conway


EXTENSIONS

159 and 175 inserted by JeanFrançois Alcover, Jun 26 2012


STATUS

approved



