A191855 -id:A191855 - cp's OEIS frontend

A007969 Rectangular numbers.

2, 5, 6, 10, 12, 13, 14, 17, 18, 20, 21, 22, 26, 28, 29, 30, 33, 34, 37, 38, 39, 41, 42, 44, 45, 46, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 82, 84, 85, 86, 89, 90, 92, 93, 94, 95, 97, 98, 101, 102, 105, 106, 108, 109

Offset: 1

J. H. Conway

nonn

A191854(n) = A007966(a(n)); A191855(n) = A007967(a(n)). - Reinhard Zumkeller, Jun 18 2011
It seems that D(n) = 4*a(n) gives precisely those even discriminants D from 4*A000037 of indefinite binary quadratic forms that have only improper solutions of the Pell equation x^2 - D*y^2 = +4. Conjecture tested for n = 1..66. Alternatively, the conjecture is that this sequence gives the r values for the Pell equation X^2 + r Y^2 = +1 whenever Y is even. See A261249 and A261250. - Wolfdieter Lang, Sep 16 2015
The proof of these two versions of the conjecture is given in the W. Lang link. - Wolfdieter Lang, Sep 19 2015 (revised Oct 03 2015)

			From _Wolfdieter Lang_, Sep 18 2015: (Start)
a(1) = 5 = 5*1 and 5*1^2 - 1*2^2  = 1.
a(7) = 14 = 2*7 and 2*2^2 - 7*1^2 = 1. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..200
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Wolfdieter Lang, Proof of a Conjecture Related to the 1-Happy Numbers.

Every number belongs to exactly one of A000290, A007969, A007970.
Cf. A191854 (B numbers), A191855 (C numbers).
Cf. A007966, A007967, A007968, A261249, A261250.
Subsequence of A000037, A002144 is a subsequence.
A263006 (R numbers), A263007 (S numbers).

a007969 n = a007969_list !! (n-1)
a007969_list = filter ((== 1) . a007968) [0..]
-- 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}]]; A007969 = Reap[Table[Print[n, " ", f[n]]; If[f[n] != {} && f[n] =!= Null, Sow[n]], {n, 1, 130}]][[2, 1]] (* Jean-François Alcover, Jun 26 2012, updated Sep 18 2015 *)

a(n) = A191854(n)*A191855(n); A007968(a(n)) = 1. - Reinhard Zumkeller, Jun 18 2011

a(n) is in the sequence if a(n) = C*B with integers B >= 1 and C >= 2, such that C*S^2 - B*R^2 = 1 has an integer solution (without loss of generality one may take S and R positive). See the Conway link. - Wolfdieter Lang, Sep 18 2015

A007967 Second factor in happy factorization of n.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 3, 1, 4, 3, 10, 11, 4, 13, 2, 5, 4, 17, 9, 19, 5, 7, 11, 1, 6, 5, 26, 27, 4, 29, 6, 1, 2, 3, 2, 7, 6, 37, 19, 13, 20, 41, 7, 43, 4, 9, 2, 1, 8, 7, 50, 51, 13, 53, 27, 5, 8, 19, 58, 59, 4, 61, 2, 9, 8, 65, 33, 67, 17, 3, 14, 1, 9, 73, 74, 3, 4, 11, 3, 1, 10, 9, 82, 83

Offset: 0

J. H. Conway

a(n) = n / A007966(n);

a(A007969(n)) = A191855(n); a(A007970(n)) = A191857(n). - Reinhard Zumkeller, Jun 18 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..300
J. H. Conway, On Happy Factorizations, J. Integer Sequences, Vol. 1, 1998, #1.
Initial Happy Factorization Data

Cf. A191914.

Cf. A007968, A007969, A007970, A191855, A191857.

import Data.List (genericIndex)
a007967 n = genericIndex a007967_list n
a007967_list = map snd hCouples
-- Pairs hCouples are defined in A007968.
-- Reinhard Zumkeller, Oct 11 2015

r[b_, c_,  d_] := (red = Reduce[x > 0 && y > 0 && b*x^2 + d == c*y^2, {x, y}, Integers] /. C[1] -> 1 // Simplify; If[Head[red] === Or, red[[1]], red]); f[n_] := f[n] =  If[IntegerQ[rn = Sqrt[n]], {0, rn, rn, rn, rn},  Catch[Do[b = bc[[1]]; c = bc[[2]]; If[c > 1 && (r0 = r[b, c, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, b, c, x0, y0}]]; If[b > 1 && (r0 = r[c, b, 1]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; Throw[{1, c, b, x0, y0}]]; If[(r0 = r[b, c, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, b, c, x0, y0}]]]; If[(r0 = r[c, b, 2]) =!= False, rr = ToRules[r0]; x0 = x /. rr; y0 = y /. rr; If[OddQ[x0] && OddQ[y0], Throw[{2, c, b, x0, y0}]]];, {bc, Union[Sort[{#, n/#}] & /@ Divisors[n]]}]]];a[n_] := f[n][[3]]; A007967 = Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 90}] (* Jean-François Alcover, Sep 18 2015 *)

A191854 First factor in happy factorization of n-th rectangular number.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 7, 1, 2, 4, 3, 2, 1, 7, 1, 5, 11, 17, 1, 2, 3, 1, 6, 11, 5, 23, 1, 4, 1, 2, 11, 7, 3, 1, 15, 1, 31, 1, 2, 4, 23, 5, 8, 1, 1, 19, 7, 26, 1, 3, 1, 2, 1, 9, 23, 3, 47, 19, 1, 49, 1, 2, 5, 1, 27, 1, 10, 3, 7, 1, 2, 4, 9, 2, 1, 31, 1, 14, 3, 1

Offset: 1

Reinhard Zumkeller, Jun 18 2011

nonn

a(n) = A007966(A007969(n)) = A007969(n) / A191855(n);
(a(n), A191855(n)) is a 1-happy couple;
notation: B 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. A007966, A007969, A191855.

a191854 = a007966 . 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}]]; A191854 = Reap[Table[Print[n, " ", f[n]]; If[f[n] != {} && f[n] =!= Null, Sow[f[n][[1]]]], {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

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)).

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

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A007969 Rectangular numbers.

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A007967 Second factor in happy factorization of n.

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A191854 First factor in happy factorization of n-th rectangular number.

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

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