A007969 -id:A007969 - cp's OEIS frontend

A191855 Second factor in happy factorization of n-th rectangular number.

2, 5, 3, 10, 4, 13, 2, 17, 9, 5, 7, 11, 26, 4, 29, 6, 3, 2, 37, 19, 13, 41, 7, 4, 9, 2, 50, 13, 53, 27, 5, 8, 19, 58, 4, 61, 2, 65, 33, 17, 3, 14, 9, 73, 74, 4, 11, 3, 82, 28, 85, 43, 89, 10, 4, 31, 2, 5, 97, 2, 101, 51, 21, 106, 4, 109, 11, 37, 16, 113, 57

Offset: 1

Reinhard Zumkeller, Jun 18 2011

nonn

a(n) > 1;
a(n) = A007967(A007969(n)) = A007969(n) / A191854(n);
(A191854(n), a(n)) is a 1-happy couple;
notation: C in the Conway link.

Reinhard Zumkeller, Table of n, a(n) for n = 1..200
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.

Cf. A007967, A007969, A191854.

a191855 = a007967 . a007969  -- 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}]]; A191855 = Reap[Table[Print[n, " ", f[n]];    If[f[n] != {} && f[n] =!= Null, Sow[f[n][[2]]]], {n, 1, 130}]][[2, 1]] (* Jean-François Alcover, Sep 18 2015 *)

Wrong formula removed (thanks to Wolfdieter Lang, who pointed this out) by Reinhard Zumkeller, Oct 11 2015

A261250 One half of the even entries of A033317.

Original entry on oeis.org

1, 2, 1, 3, 1, 90, 2, 4, 2, 1, 6, 21, 5, 12, 910, 1, 2, 3, 6, 3, 2, 160, 1, 15, 12, 1794, 7, 45, 4550, 33, 6, 1, 10, 1287, 2, 113076990, 4, 8, 4, 2, 468, 15, 1, 133500, 215, 3315, 20, 3, 9, 3, 15498, 561, 26500, 1, 60, 630, 110532, 2, 3188676, 5, 10, 5, 2, 1557945, 65, 7570212227550, 1, 14, 6, 56648, 48, 455, 30, 14127

Offset: 1

Wolfdieter Lang, Sep 16 2015

nonn

2*a(n) = y0(n) is the positive fundamental solution satisfying the Pell equation x0(n)^2 + D(n)*y0(n)^2 = +1 with D(n) coinciding apparently with Conway's rectangular numbers r(n) = A007969(n). The corresponding x0 values are given in A262024.

For a proof of this coincidence see the W. Lang link under A007969. - Wolfdieter Lang, Oct 04 2015

			The [r(n), x0(n), y0(n)] values for n = 1..16 are:
[2, 3, 2], [5, 9, 4], [6, 5, 2], [10, 19, 6],
[12, 7, 2], [13, 649, 180], [14, 15, 4],
[17, 33, 8], [18, 17, 4], [20, 9, 2],
[21, 55, 12], [22, 197, 42], [26, 51, 10],
[28, 127, 24], [29, 9801, 1820], [30, 11, 2], ...

Cf. A033317, A033313, A000037, A007969, A262024, A262025.

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2 n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
Select[DeleteCases[PellSolve /@ Range[200], {}][[All, 2]], EvenQ]/2 (* Jean-François Alcover, Aug 12 2023, using the PellSolve code given in A033317 *)

A191860 First member of a pair of numbers occurring in the definition of 1-happy couples.

Original entry on oeis.org

1, 2, 2, 3, 3, 1, 18, 4, 4, 13, 1, 3, 5, 5, 3, 70, 1, 1, 6, 6, 3, 3, 32, 59, 3, 4, 7, 7, 9, 182, 11, 2, 1, 5, 23, 1, 29718, 8, 8, 221, 2, 13, 7, 1, 1068, 1, 39, 5, 9, 9, 3, 378, 7, 500, 11, 5, 45, 151, 1, 5604, 10, 10, 5, 2, 31, 5, 8890182, 1, 7, 3, 776, 15

Offset: 1

Reinhard Zumkeller, Jun 18 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..200
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Initial Happy Factorization Data

Cf. A007968, A007969, A094048 (subsequence).

a191860 = fst . sqrtPair . a007969
Function sqrtPair is defined in A007968.
-- Reinhard Zumkeller, Oct 11 2015

Wrong comment and wrong formula removed (thanks to Wolfdieter Lang, who pointed this out) by Reinhard Zumkeller, Oct 11 2015

A262026 The positive odd fundamental solutions y = y0(n) for the Pell equation x^2 - d*y^2 = +1. It turns out that d = d(n) coincides with A007970(n).

Original entry on oeis.org

1, 3, 1, 3, 1, 39, 5, 1, 5, 273, 3, 1, 3, 531, 7, 1, 7, 69, 1, 5967, 413, 3, 9, 1, 9, 3, 21, 165, 5, 1, 22419, 5, 93, 105, 11, 1, 11, 419775, 51, 927, 21, 3, 6578829, 1, 140634693, 3, 105, 57, 5019135, 13, 1, 13, 153, 15, 313191, 123, 650783, 7, 1, 1153080099, 7, 45, 19162705353, 3, 33, 5

Offset: 1

Wolfdieter Lang, Oct 04 2015

nonn

The corresponding x = x0(n) values are given by A262027(n).
This is a proper subset of A033317 corresponding to its odd members.
For the proof that d(n) = A007970(n), the products of Conway's 2-happy couples, see the W. Lang link under A007970.
For the positive even fundamental solutions y = y0(n) of x^2 - d*y^2 = 1, where d = d(n) coincides with A007969(n) see 2*A261250(n).
If d(n) = A007970(n) is odd (necessarily congruent to 3 modulus 4) then x0(n) is even, and if d(n) is even (necessarily congruent to 0 modulus 8) then x0 is odd.

			The first triples [d(n), x0(n), y0(n)] are: [3,2,1], [7,8,3], [8,3,1], [11,10,3], [15,4,1], [19,170,39], [23,24,5], [24,5,1], [27,26,5], [31,1520,273], [32,17,3], [35,6,1], [40,19,3], [43,3482,531], [47,48,7], [48,7,1], [51,50,7], [59,530,69], [63,8,1], [67,48842,5967], [71,3480,413], [75,26,3], [79,80,9], [80,9,1], [83,82,9], [87,28,3], [88,197,21], [91,1574,165], [96,49,5], [99,10,1], [103,227528,22419], ...

Cf. A007970, A033317, A262027, A262028.

x0(n)^2 - d(n)*a(n)^2 = +1 with x0(n) =

A262027(n) and d(n) = A007970(n). (x0(n), y0(n) = a(n)) are the positive fundamental solutions of this Pell equation x^2 - d*y^2 = +1 with odd y = y0.

A262024 Positive fundamental solution x0 corresponding to the even y0 = 2*A261250 of the Pell equation x^2 - D y^2 = +1.

Original entry on oeis.org

3, 9, 5, 19, 7, 649, 15, 33, 17, 9, 55, 197, 51, 127, 9801, 11, 23, 35, 73, 37, 25, 2049, 13, 199, 161, 24335, 99, 649, 66249, 485, 89, 15, 151, 19603, 31, 1766319049, 63, 129, 65, 33, 7775, 251, 17, 2281249, 3699, 57799, 351, 53, 163, 55, 285769, 10405, 500001, 19, 1151, 12151, 2143295, 39, 62809633, 99, 201, 101, 41, 32080051, 1351, 158070671986249, 21, 295, 127, 1204353, 1025, 9801, 649, 306917

Offset: 1

Wolfdieter Lang, Sep 16 2015

nonn

This is a proper subset of A033313 corresponding to the even members of A033317.
The D values coincide apparently with A007969 (Conway's rectangular numbers).
For a proof of this coincidence see the W. Lang link under A007969. - Wolfdieter Lang, Oct 04 2015

			See A261250.

Cf. A000037, A033313, A033317, A261250, A007969.

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)).

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

Wolfdieter Lang, Oct 28 2015

nonn

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.

			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).

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

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

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)).

A191861 Second member of a pair of numbers occurring in the definition of 1-happy couples.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 4, 1, 3, 1, 2, 5, 1, 4, 13, 3, 2, 6, 1, 1, 1, 5, 9, 5, 3, 7, 1, 5, 25, 3, 3, 1, 2, 3, 2, 3805, 8, 1, 27, 1, 36, 59, 1, 125, 3, 85, 4, 9, 1, 1, 41, 3, 53, 15, 12, 14, 732, 5, 569, 10, 1, 1, 1, 3, 13, 851525, 1, 2, 2, 73, 7, 13, 5, 11, 1

Offset: 1

Reinhard Zumkeller, Jun 18 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..200
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Initial Happy Factorization Data

Cf. A007968, A007969.

a191861 = snd . sqrtPair . a007969
-- Function sqrtPair is defined in A007968.
-- Reinhard Zumkeller, Oct 11 2015

Wrong comment and wrong formula removed (thanks to Wolfdieter Lang, who pointed this out) by Reinhard Zumkeller, Oct 11 2015

A261249 Number of classes of proper solutions of the Pell equation x^2 - D(n) y^2 = +4 for D(n) = A079896(n), n >= 1.

Original entry on oeis.org

2, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Wolfdieter Lang, Sep 16 2015

nonn

See the W. Lang link on A225953, Table 2. References will also be found there. For the present class number see especially Theorem 109 pp. 207-208 of the Nagell reference.
These class numbers should not be confused with the class numbers of indefinite binary quadratic forms of discriminant D(n), which are given in A087048(n).
If a(n) = 2 then the proper positive fundamental solution for the second class [x2(n), y2(n)] is obtained from the solution of the first class [x1(n), y1(n)] (shown in the mentioned Table 2 under Pell(X, Y)) by application of the matrix M(n) = [[x0(n), D(n)*y0(n)], [y0(n), x0(n)]] on (x1(n), -y1(n))^T (T for transposed), where x0(n) and y0(n) is the positive (proper) fundamental solution of x^2 - D(n)*y^2 = +1 found under A033313 and A033317 for the appropriate D from A000037. Application of positive powers of M(n) to the proper positive fundamental solution of each class produces all positive solutions.
If a(n) = 1 the class is called ambiguous (see Nagell, p. 205). In this case the proper positive fundamental solution [x1(n), y1(n)] = [x(n), y(n)] and the negative one [x1(n), -y1(n)] belong to the same class.
For every D(n) = A079896(n) there is the improper positive fundamental solution [2*x0(n), 2*y0(n)].
Conjecture: For even D(n), i.e., D from 4*A000037, and a(n) = 0 one finds for r(n) = D(n)/4 coincidence with Conway's so-called rectangular numbers A007969. The first D values are 8, 20, 24, 40, 48, 52, 56, 68, 72, 80, ... This is equivalent to the conjecture that X^2 - r*y^2 = +1 has an even fundamental positive solution y = y0 precisely for the numbers A007969 (because x has to be even, x = 2*X, and whenever y0 is even all y solutions are even). See A261250 and A262024 for the y0 and x0 values, respectively.

			n=1: D(1) = 5 = A000037(3) with the a(1) = 2 proper positive fundamental solutions [x, y] = [3, 1] and [7, 3] for the two classes.
  [x0(1), y0(1)] = [A033313(3), A033317(3)] = [9, 4], and (7, 3)^T = [[9, 4*5], [4, 9]] (3, -1)^T.
  All other positive solutions in each of the two classes are obtained by applying positive powers of this matrix M(5) to the fundamental solutions.
  The improper positive fundamental solution is [2*9, 2*4] = [18, 8].
n=2: D(2) = 8 = A000037(6) has a(2) = 0, hence there are only the improper solutions obtainable from [2*3, 2*1] = [6, 2], the smallest positive one. For this even D one has, with x = 2*X, X^2 - 8/4 y^2 = +1, which has an even positive fundamental solution y0 = 2, and r(2) = D(2)/4 = 2 is A007969(1).

Nagell, T. Introduction to number theory, Chelsea Publishing Company, 1964, page 52.

Cf. A079896, A225953, A087048, A033313, A033317, A261250, A262024.

Offset corrected by Robin Visser, Jun 08 2025

A262324 Conway's triangle of "happy factorizations" (flattened).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 3, 2, 2, 1, 5, 2, 3, 7, 1, 2, 4, 3, 3, 1, 10, 1, 11, 3, 4, 1, 13, 7, 2, 3, 5, 4, 4, 1, 17, 2, 9, 1, 19, 4, 5, 3, 7, 2, 11, 23, 1, 4, 6, 5, 5, 1, 26, 1, 27, 7, 4, 1, 29, 5, 6, 31, 1, 16, 2, 11, 3, 17, 2, 5, 7, 6, 6, 1, 37, 2, 19, 3, 13, 2, 20, 1, 41, 6, 7, 1, 43, 11, 4, 5, 9, 23, 2, 47, 1, 6, 8, 7, 7

Offset: 0

Jean-François Alcover, Sep 18 2015

Conway's triangle is listed by increasing couple products, with duplicate squares removed.

			Triangle begins:
{0,0},
{1,1},
{1,2},   {1,3},  {2,2},
{1,5},   {2,3},  {7,1},  {2,4}, {3,3},
{1,10}, {1,11},  {3,4}, {1,13}, {7,2},  {3,5},  {4,4},
{1,17},  {2,9}, {1,19},  {4,5}, {3,7}, {2,11}, {23,1}, {4,6}, {5,5},
...
The original triangle (adapted and truncated):
                           ...
                      5^2  ...
                 4^2  1*26 ...
            3^2  1*17 1*27 ...
        2^2 1*10 2*9  7*4  ...
    1^2 1*5 1*11 1*19 1*29 ...
0^2 1*2 2*3 3*4  4*5  5*6  ...
1^2 1*3 7*1 1*13 3*7  31*1 ...
    2^2 2*4 7*2  2*11 16*2 ...
        3^2 3*5  23*1 11*3 ...
            4^2  4*6  17*2 ...
                 5^2  5*7  ...
                      6^2  ...
                           ...

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

Cf. A007966, A007969, A007970, A191854, A191855, A191856.

f[0] = {0, 0}; f[32] = {16, 2}(* to speed up *); f[n_] := Do[c = n/b; If[b == c, Return[{b, b}]]; r1 = Reduce[r >= 0 && s >= 0 && c > 1 && b*r^2 + 1 == c*s^2, {r, s}, Integers]; If[r1 =!= False, Return[{b, c}]]; r2 = Reduce[r >= 0 && s >= 0 && r == 2x + 1 && s == 2y + 1 && b*r^2 + 2 == c *s^2, {r, s, x, y}, Integers]; If[r2 =!= False, Return[{b, c}]], {b, Divisors[n]}]; Table[Print["f(", n, ") = ", fn = f[n]]; fn, {n, 0, 49}] // Flatten

cp's OEIS Frontend

A191855 Second factor in happy factorization of n-th rectangular number.

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A261250 One half of the even entries of A033317.

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A191860 First member of a pair of numbers occurring in the definition of 1-happy couples.

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A262026 The positive odd fundamental solutions y = y0(n) for the Pell equation x^2 - d*y^2 = +1. It turns out that d = d(n) coincides with A007970(n).

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A262024 Positive fundamental solution x0 corresponding to the even y0 = 2*A261250 of the Pell equation x^2 - D y^2 = +1.

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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)).

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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)).

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A191861 Second member of a pair of numbers occurring in the definition of 1-happy couples.

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A261249 Number of classes of proper solutions of the Pell equation x^2 - D(n) y^2 = +4 for D(n) = A079896(n), n >= 1.

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