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 1-5 of 5 results.

A094048 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least m for which m^2+1=p(n)*k^2 has a solution.

Original entry on oeis.org

2, 18, 4, 70, 6, 32, 182, 29718, 1068, 500, 5604, 10, 8890182, 776, 1744, 113582, 4832118, 1118, 1111225770, 1764132, 14, 1710, 23156, 71011068, 16, 82, 8920484118, 1063532, 2482, 126862368, 352618
Offset: 1

Views

Author

Matthijs Coster, Apr 29 2004

Keywords

Comments

Subsequence of A191860. [Reinhard Zumkeller, Jun 18 2011]

Crossrefs

Cf. A002144, A094049 (associated k), A130226, A137351, A179073.
Cf. A000196, A037213, A000290.

Programs

  • Haskell
    a094048 n = head [m | m <- map (a037213 . subtract 1 . (* a002144 n))
                               (tail a000290_list), m > 0]
-- Reinhard Zumkeller, Jun 13 2015
  • Mathematica
    f[n_] := Block[{y = 1}, While[ !IntegerQ[ Sqrt[n*y^2 - 1]], y++ ]; Sqrt[n*y^2 - 1]]; lst = {}; Do[p = Prime@ n; If[ Mod[p, 4] == 1, AppendTo[lst, f@p]; Print[{n, Prime@n, f@p}]], {n, 66}]; lst

Extensions

Edited by Don Reble, Apr 30 2004

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

Views

Author

Wolfdieter Lang, Oct 28 2015

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

Cf. A007969, A191854, A191855, A191860, A263006, A262025, A261250.

Formula

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)).

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)).

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)).

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 1-5 of 5 results.

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