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.

Previous Showing 11-18 of 18 results.

A262025 a(n) = (A262024(n)-1)/2: a(n)*(a(n) + 1) = d(n)*Y(n)^2 with d(n) = A007969 and Y(n) = A261250(n).

Original entry on oeis.org

1, 4, 2, 9, 3, 324, 7, 16, 8, 4, 27, 98, 25, 63, 4900, 5, 11, 17, 36, 18, 12, 1024, 6, 99, 80, 12167, 49, 324, 33124, 242, 44, 7, 75, 9801, 15, 883159524, 31, 64, 32, 16, 3887, 125, 8, 1140624, 1849, 28899, 175, 26, 81, 27, 142884, 5202, 250000, 9, 575, 6075, 1071647, 19, 31404816, 49, 100, 50, 20, 16040025, 675, 79035335993124, 10, 147, 63, 602176, 512, 4900, 324, 153458
Offset: 1

Views

Author

Wolfdieter Lang, Sep 19 2015

Keywords

Comments

The positive fundamental solutions (x0(n), y0(n)) of the Pell equation x^2 - d(n) y^2 = +1, with d not a square, have only even y solutions for d(n) = A007969 (Conway's products of 1-happy couples). The proof is now given in the W. Lang link under A007969. The solutions x0 and y0 = 2*Y0 are given in A262024 and 2*A261250, respectively. The numbers X0(n) = (x0(n) - 1)/2 = a(n) satisfy a(n)*(a(n) + 1) = d(n)*Y0(n)^2. See the mentioned link.

Crossrefs

Cf. A007969, A007970, A262024, A261250.

A262028 a(n) = (A262026(n) - 1)/2.

Original entry on oeis.org

0, 1, 0, 1, 0, 19, 2, 0, 2, 136, 1, 0, 1, 265, 3, 0, 3, 34, 0, 2983, 206, 1, 4, 0, 4, 1, 10, 82, 2, 0, 11209, 2, 46, 52, 5, 0, 5, 209887, 25, 463, 10, 1, 3289414, 0, 70317346, 1, 52, 28, 2509567, 6, 0, 6, 76, 7, 156595
Offset: 1

Views

Author

Wolfdieter Lang, Oct 04 2015

Keywords

Comments

This is the column Y_0 of the Table of a proof given as a W. Lang link under A007970.
(x0(n), y0(n) = 2*a(n) + 1) with x0(n) = A262067(n) are the fundamental solutions of the Pell equation x^2 - d*y^2 = +1 with odd y. The d values coincide with d = d(n) = A007970(n). For a proof see the mentioned link.

Examples

			For the first triples [d(n), x0(n), 2*a(n) + 1] see A262066.

Crossrefs

Cf. A006970, A262026, A262067.

Formula

A262067(n)^2 - A007970(n)*(2*a(n) + 1)^2 = +1, n >= 1.

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

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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)).

A263009 Second member U0(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, 3, 1, 1, 1, 3, 5, 1, 1, 39, 3, 1, 1, 9, 7, 1, 1, 3, 1, 27, 59, 3, 9, 1, 1, 1, 3, 15, 5, 1, 477, 1, 3, 7, 11, 1, 1, 2175, 17, 9, 7, 3, 747, 1, 41571, 1, 5, 19, 627, 13, 1, 1, 9, 5, 153
Offset: 1

Views

Author

Wolfdieter Lang, Oct 29 2015

Keywords

Comments

See A263008. E(n)*a(n)^2 - D(n)*A263008(n)^2 = +2, n >= 1, with the 2-happy couple (D(n), E(n)) = (A191856(n), A191857(n)). The 2-happy numbers D(n)*E(n) are given by A007970(n).
In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = U0(n) numbers appear for the t = 2 rows in column w.

Examples

			n = 4: 2-happy number A007970(4) = 11 = 1*11 =
  A191856(4)*A191857(4). 11*a(4)^2 - 1*A263008(4)^2 = 11*1^2 - 1*3^2 = +2. This is the smallest positive solution for given (D, E) = (1, 11).

Links

Crossrefs

Cf. A007970, A191856, A191857, A191860, A263008, A262026 , A262027, A262028.

Formula

A191857(n)*a(n)^2 - A191856(n)*A263008(n)^2 = +2, and A263008(n) with a(n) is the smallest positive
solution for the given 1-happy couple (A191856(n), A191857(n)).

A191862 First member of a pair of numbers occurring in the definition of 2-happy couples.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 7, 1, 5, 1, 7, 1, 2, 1, 23, 1, 7, 99, 1, 4, 5, 43, 1, 1, 9, 51, 3, 1, 1, 1, 5, 47, 4005, 16, 277, 1, 11, 4, 193, 57, 191, 3, 1, 1, 2, 3383, 7, 21, 70, 20621, 1, 13, 5, 17, 20, 25, 51, 217, 1, 7, 9041, 5, 1, 416941, 1, 251, 1, 1, 15, 3, 5
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 18 2011

Keywords

Links

Crossrefs

Cf. A007968, A007970.

Programs

Extensions

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

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)).

A191863 Second member of a pair of numbers occurring in the definition of 2-happy couples.

Original entry on oeis.org

1, 1, 3, 1, 2, 1, 3, 5, 1, 1, 39, 3, 1, 1, 78, 7, 1, 13, 4, 1, 3, 5, 3, 9, 1, 11, 1, 1, 2, 5, 1, 477, 389, 3, 51, 11, 1, 5, 2175, 5, 33, 7, 2, 6, 1, 41571, 44, 5, 11, 3201, 13, 1, 11, 9, 3, 11, 37, 2999, 7, 1, 127539, 8, 1, 57003, 3, 17, 5, 15, 1, 1, 1, 17
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 18 2011

Keywords

Comments

a(n) is odd, as is A191862(n).

Links

Crossrefs

Cf. A007968, A007970.

Programs

Extensions

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

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

Views

Author

Jean-François Alcover, Sep 18 2015

Keywords

Comments

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

Examples

			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  ...
                           ...

Links

Crossrefs

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

Programs

  • Mathematica
    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
Previous Showing 11-18 of 18 results.

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