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A007970 Rhombic numbers. 19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
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
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 *)
PROG
(Haskell)
a007970 n = a007970_list !! (n-1)
a007970_list = filter ((== 2) . a007968) [0..]
-- Reinhard Zumkeller, Oct 11 2015
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).
Sequence in context: A047528 A069122 A278519 * A255342 A332572 A134258
KEYWORD
nonn
AUTHOR
EXTENSIONS
159 and 175 inserted by Jean-François Alcover, Jun 26 2012
STATUS
approved

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Last modified February 27 11:45 EST 2024. Contains 370378 sequences. (Running on oeis4.)