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.

A198409 Positions in sequences A198384, A198385 and A198386 to indicate triples of squares in arithmetic progression, that are not multiples of earlier triples.

Original entry on oeis.org

1, 3, 5, 7, 10, 13, 15, 23, 24, 26, 30, 35, 39, 42, 45, 47, 51, 54, 62, 69, 70, 72, 83, 84, 88, 97, 98, 102, 107, 114, 115, 124, 126, 129, 136, 141, 142, 143, 156, 157, 167, 169, 172, 177, 181, 188, 191, 201, 205, 208, 214, 218, 229, 230, 237, 244, 249, 253
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Comments

A198435(n) = A198384(a(n)); A198439(n) = A198388(a(n));
A198436(n) = A198385(a(n)); A198440(n) = A198389(a(n));
A198437(n) = A198386(a(n)); A198441(n) = A198390(a(n));
A198438(n) = A198387(a(n)).

Links

Programs

  • Haskell
    import Data.List (elemIndices)
a198409 n = a198409_list !! (n-1)
a198409_list = map (+ 1) $ elemIndices 1 $ map a008966 $
   zipWith gcd a198384_list $ zipWith gcd a198385_list a198386_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
Position[tt, t_List /; SquareFreeQ[GCD@@t]] // Flatten (* Jean-François Alcover, Oct 24 2021 *)

Formula

A008966(A050873(A198439(n),A050873(A198440(n),A050873(A198441(n))))) = 1.

A198435 First term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384,A198385,A198386).

Original entry on oeis.org

1, 49, 49, 289, 1, 529, 961, 2401, 289, 2209, 529, 5041, 49, 1681, 1681, 6241, 9409, 49, 961, 5329, 16129, 14161, 7921, 289, 25921, 2209, 12769, 27889, 14161, 1, 39601, 2401, 5329, 10609, 25921, 49729, 58081, 529, 961, 10609, 7921, 36481, 82369, 22801, 47089
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Links

Crossrefs

Cf. A198384, A198409, A198437, A198438, A198439

Programs

  • Haskell
    a198435 n = a198435_list !! (n-1)
a198435_list = map a198384 a198409_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u^2, v^2, w^2}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD @@ t>1 && MemberQ[tt, t/GCD @@ t]][[All, 1]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

a(n) = A198439(n)^2 = A198384(A198409(n));
A198436(n) - a(n) = A198437(n) - A198436(n) = A198438(n).

A198390 Square root of third term of a triple of squares in arithmetic progression.

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 98, 102, 103, 105, 112, 113, 115, 119, 119, 119, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Comments

There is a connection to the leg sums of Pythagorean triangles.
See a comment on the primitive case under A198439, which applies mutatis mutandis. - Wolfdieter Lang, May 23 2013
Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013
n appears A331671(n) times. - Ray Chandler, Feb 26 2020

Examples

			Connection to leg sums of Pythagorean triangles: a(2) = 14 because (in the notation of the Zumkeller link) (u,v,w)= (2,10,14) = 2*(1,5,7), and this corresponds to the non-primitive Pythagorean triangle 2*(x=(7-1)/1,y=(1+7)/2,z=5) = 2*(3,4,5) with leg sum 2*(3+4) = 14. - _Wolfdieter Lang_, May 23 2013

Links

Crossrefs

Cf. A198386, A198409, A198439, A198441, A331671.

Programs

  • Haskell
    a198390 n = a198390_list !! (n-1)
a198390_list = map (\(,,x) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 3]] (* Jean-François Alcover, Oct 20 2021 *)
  • PARI
    is(n)=my(t=n^2);forstep(i=2-n%2,n-2,2, if(issquare((t+i^2)/2), return(1))); 0 \\ Charles R Greathouse IV, May 28 2013

Formula

A198386(n) = a(n)^2.
A198441(n) = a(A198409(n)).

A198388 Square root of first term of a triple of squares in arithmetic progression.

Original entry on oeis.org

1, 2, 7, 3, 7, 4, 17, 14, 5, 1, 6, 14, 23, 7, 31, 21, 8, 34, 9, 28, 21, 10, 49, 17, 11, 47, 2, 12, 35, 23, 13, 28, 51, 46, 71, 14, 62, 42, 7, 15, 16, 41, 35, 17, 41, 49, 79, 3, 68, 18, 97, 19, 56, 7, 42, 20, 69, 98, 34, 21, 93, 31, 63, 22, 85, 94, 23, 49, 73
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

Comments

There is a connection to |x-y| of Pythagorean triangles (x,y,z). See a comment on the primitive Pythagorean triangle case under A198441 which applies mutatis mutandis. - Wolfdieter Lang, May 23 2013

Examples

			Connection to Pythagorean triangles: a(2) = 2 because (in the notation of the Zumkeller link) (u,v,w) = 2*(1,5,7) and the corresponding Pythagorean triangle is 2*((7-1)/2,(1+7)/2,5) = 2*(3,4,5) with |x-y| = 2*(4-3) = 2. - _Wolfdieter Lang_, May 23 2013

Links

Crossrefs

Cf. A198384, A198409, A198439.

Programs

  • Haskell
    a198388 n = a198388_list !! (n-1)
a198388_list = map (\(x,,) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

A198384(n) = a(n)^2.
A198439(n) = a(A198409(n)).

A198440 Square root of second term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 61, 53, 65, 65, 85, 73, 85, 89, 101, 113, 97, 109, 125, 145, 145, 149, 137, 181, 157, 173, 197, 185, 169, 221, 185, 193, 205, 229, 257, 265, 205, 221, 233, 241, 269, 313, 265, 293, 325, 277, 317, 281, 365, 289, 305, 305, 365, 401
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

Comments

This sequence gives the hypotenuses of primitive Pythagorean triangles (with multiplicities) ordered according to nondecreasing values of the leg sums x+y (called w in the Zumkeller link, given by A198441). See the comment on the equivalence to primitive Pythagorean triangles in A198441. For the values of these hypotenuses ordered nondecreasingly see A020882. See also the triangle version A222946. - Wolfdieter Lang, May 23 2013

Examples

			From _Wolfdieter Lang_, May 22 2013: (Start)
Primitive Pythagorean triangle (x,y,z), even y, connection:
a(8) = 61 because the leg sum x+y = A198441(8) = 71 and due to A198439(8) = 49 one has y = (71+49)/2 = 60 is even, hence x = (71-49)/2 = 11 and z = sqrt(11^2 + 60^2) = 61. (End)

Links

Crossrefs

Cf. A020882, A222946.

Programs

  • Haskell
    a198440 n = a198440_list !! (n-1)
a198440_list = map a198389 a198409_list
  • Mathematica
    wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]][[All, 2]] (* Jean-François Alcover, Oct 22 2021 *)

Formula

A198436(n) = a(n)^2; a(n) = A198389(A198409(n)).
Showing 1-5 of 5 results.

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