cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A007970 Rhombic numbers.

Original entry on oeis.org

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

Views

Author

J. H. Conway

Keywords

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

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

A263008 First member T0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 13, 1, 1, 5, 7, 1, 1, 3, 59, 1, 1, 7, 23, 1, 221, 7, 1, 1, 1, 9, 3, 7, 11, 1, 1, 47, 5, 31, 15, 1, 1, 11, 193, 3, 103, 3, 1, 8807, 1, 3383, 3, 21, 3, 8005, 1, 1, 13, 17, 3, 2047
Offset: 1

Views

Author

Wolfdieter Lang, Oct 29 2015

Keywords

Comments

The 2-happy numbers D(n)*E(n) are given in A007970(n) (called rhombic numbers in the Conway paper). D(n) = A191856(n), E(n) = A191857(n). Here the corresponding smallest positive numbers satisfying E(n)*U(n)^2 - D(n)*T(n)^2 = +2, n >= 1, with odd U(n) and T(n) are given as T0(n) = a(n) and U0(n) = A263009(n).
In the W. Lang link the first U0(n) and T0(n) numbers are given in the Table for d(n) = A007970(n), n >= 1.
In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = T0(n) numbers appear for the t = 2 rows in column v.

Examples

			n = 6: 2-happy number A007970(6) = 19 = 1*19 = A191856(6)*A191857(6). 19*A263009(6)^2 - 1*a(6)^2 = 19*3^2 - 1*13^2 = +2. This is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).

Links

Crossrefs

Cf. A007970, A191856, A191857, A191860, A263009, A262026, A262027, A262028.

Formula

A191857(n)*A263009(n)^2 - A191856(n)*a(n)^2 = +2, and a(n) with A263009(n) is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).

A263006 First member R0(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, 2, 1, 3, 1, 18, 1, 4, 2, 1, 3, 7, 5, 3, 70, 1, 1, 1, 6, 3, 2, 32, 1, 3, 4, 23, 7, 9, 182, 11, 2, 1, 5, 99, 1, 29718, 1, 8, 4, 2, 13, 5, 1, 1068, 43, 39, 5, 1, 9, 3, 378, 51, 500, 1, 5, 45, 151, 1, 5604, 1, 10, 5, 2, 4005, 5, 8890182, 1, 7, 3, 776, 16, 35, 6, 277
Offset: 1

Views

Author

Wolfdieter Lang, Oct 28 2015

Keywords

Comments

The 1-happy numbers B(n)*C(n) are given in A007969(n) (called rectangular numbers in the Conway paper). B(n) = A191854(n), C(n) = A191855(n). Here the corresponding smallest positive numbers satisfying C(n)*S0(n)^2 - B(n)*R0(n)^2 = +1, n >= 1, are given as R0(n) = a(n) and S0(n) = A263007(n).
For a proof of Conway's happy number factorization theorem see the W. Lang link under A007970.
In the W. Lang link given in A007969 the first C(n), B(n), S0(n), R0(n) numbers are given in the Table for d(n) = A007969(n), n >= 1.
In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = R0(n) numbers appear for the t = 1 rows in column v.

Examples

			n = 6: 1-happy number A007969(6) = 13 = 1*13 = A191854(6)*A191855(6). 13*A263007(6)^2 - 1*a(6)^2 = 13*5^2 - 1*18^2 = +1. This is the smallest positive solution for (B, C) = (1, 13).

Links

Crossrefs

Cf. A007969, A007970, A191854, A191855, A191860, A263007, A262025, A261250, A263008, A263009.

Formula

A191855(n)*A263007(n)^2 - A191854(n)*a(n)^2 = +1, and a(n) with A263007(n) is the smallest positive solution for the given 1-happy couple (A191854(n), A191855(n)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.