A198436 -id:A198436 - cp's OEIS frontend

A198385 Second of a triple of squares in arithmetic progression.

25, 100, 169, 225, 289, 400, 625, 676, 625, 841, 900, 1156, 1369, 1225, 1681, 1521, 1600, 2500, 2025, 2704, 2601, 2500, 3721, 2809, 3025, 4225, 3364, 3600, 4225, 4225, 4225, 4624, 5625, 5476, 7225, 4900, 6724, 6084, 5329, 5625, 6400, 7225, 7225, 7225, 7921

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

a198385 n = a198385_list !! (n-1)
a198385_list = map (^ 2) a198389_list

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]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 2]] (* Jean-François Alcover, Oct 19 2021 *)

a(n) = A198389(n)^2.
a(n) - A198384(n) = A198386(n) - a(n) = A198387(n).
A198436(n) = a(A198409(n)).

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

Reinhard Zumkeller, Oct 25 2011

nonn

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)).

Ray Chandler, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Table of initial values

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

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 *)

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

Reinhard Zumkeller, Oct 25 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A198384, A198409, A198437, A198438, A198439

a198435 n = a198435_list !! (n-1)
a198435_list = map a198384 a198409_list

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 *)

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

A198436(n) - a(n) = A198437(n) - A198436(n) = A198438(n).

A198437 Third 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

49, 289, 529, 961, 1681, 2209, 2401, 5041, 5329, 6241, 7921, 9409, 10609, 12769, 14161, 14161, 16129, 18769, 22801, 25921, 25921, 27889, 36481, 37249, 39601, 47089, 47089, 49729, 54289, 57121, 58081, 66049, 69169, 73441, 78961, 82369, 82369, 83521, 96721

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A198435, A198436, A198438, A198409, A198441.

a198437 n = a198437_list !! (n-1)
a198437_list = map a198386 a198409_list

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, 3]] (* Jean-François Alcover, Oct 20 2021 *)

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

a(n) - A198436(n) = A198436(n) - A198435(n) = A198438(n).

A198438 Common differences in triples of squares in arithmetic progression, that are not a multiples of other triples in (A198384, A198385, A198386).

Original entry on oeis.org

24, 120, 240, 336, 840, 840, 720, 1320, 2520, 2016, 3696, 2184, 5280, 5544, 6240, 3960, 3360, 9360, 10920, 10296, 4896, 6864, 14280, 18480, 6840, 22440, 17160, 10920, 20064, 28560, 9240, 31824, 31920, 31416, 26520, 16320, 12144, 41496, 47880, 43680, 50160

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

a198438 n = a198438_list !! (n-1)
a198438_list = map a198387 a198409_list

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];
#[[2]] - #[[1]]& /@ DeleteCases[tt, t_List /; GCD@@t > 1 && MemberQ[tt, t/GCD@@t]] (* Jean-François Alcover, Oct 22 2021 *)

a(n) = A198387(A198409(n)) = A198436(n) - A198435(n) = A198437(n) - A198436(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

Reinhard Zumkeller, Oct 25 2011

nonn

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

			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)

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A020882, A222946.

a198440 n = a198440_list !! (n-1)
a198440_list = map a198389 a198409_list

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 *)

A198436(n) = a(n)^2; a(n) = A198389(A198409(n)).

A360280 Squares that are the hypotenuse of a primitive Pythagorean triangle.

Original entry on oeis.org

25, 169, 289, 625, 841, 1369, 1681, 2809, 3721, 4225, 5329, 7225, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 21025, 22201, 24649, 28561, 29929, 32761, 34225, 37249, 38809, 42025, 48841, 52441, 54289, 58081, 66049, 70225, 72361, 76729, 78961, 83521, 85849, 93025, 97969

Offset: 1

Alexandru Petrescu, Feb 01 2023

nonn

All terms are congruent to 1 (mod 8).

			169 is a term because 169 = 13^2 and (119,120,169) is a primitive Pythagorean triangle.

Cf. A000290, A008846, A020882, A198436.

a(n) = A008846(n)^2.

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A198385 Second of a triple of squares in arithmetic progression.

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

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

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A198437 Third term of a triple of squares in arithmetic progression, which is not a multiple of another triple in (A198384,A198385,A198386).

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Haskell

Mathematica

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A198438 Common differences in triples of squares in arithmetic progression, that are not a multiples of other triples in (A198384, A198385, A198386).

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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).

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Crossrefs

Programs

Haskell

Mathematica

Formula

A360280 Squares that are the hypotenuse of a primitive Pythagorean triangle.

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Author

Keywords

Comments

Examples

Crossrefs

Formula