A198437 -id:A198437 - cp's OEIS frontend

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

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.

A198386 Third of a triple of squares in arithmetic progression.

Original entry on oeis.org

49, 196, 289, 441, 529, 784, 961, 1156, 1225, 1681, 1764, 2116, 2209, 2401, 2401, 2601, 3136, 3844, 3969, 4624, 4761, 4900, 5041, 5329, 5929, 6241, 6724, 7056, 7225, 7921, 8281, 8464, 8649, 8836, 9409, 9604, 9604, 10404, 10609, 11025, 12544, 12769, 13225

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

a198386 n = a198386_list !! (n-1)
a198386_list = map (^ 2) a198390_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, 3]] (* Jean-François Alcover, Oct 19 2021 *)

a(n) = A198390(n)^2.
a(n) - A198385(n) = A198385(n) - A198384(n) = A198387(n).
A198437(n) = a(A198409(n)).

A198441 Square root of third 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

7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 119, 127, 137, 151, 161, 161, 167, 191, 193, 199, 217, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 287, 289, 311, 313, 329, 329, 337, 343, 353, 359, 367, 383, 391, 391, 401, 409, 431, 433

Offset: 1

Reinhard Zumkeller, Oct 25 2011

nonn

This sequence gives the sum of the two legs (catheti) x + y of primitive Pythagorean triangles (x,y,z) with y even and gcd(x,y) = 1, ordered nondecreasingly (with multiple entries). See A058529(n), n>=2, for the sequence without multiple entries. For the proof, put in the Zumkeller link w = x + y, v = z and u = abs(x - y). This works because w^2 - v^2 = v^2 - u^2, hence u^2 = 2*v^2 - w^2 = 2*z^2 - (x+y)^2 = 2*(x^2 + y^2) - (x+y)^2 = x^2 + y^2 - 2*x*y = (x-y)^2. The primitivity of the arithmetic progression triples follows from the one of the Pythagorean triples: gcd(u,w) = 1 follows from gcd(x,y) = 1, then gcd(u,v,w) = gcd(gcd(u,w),v) = 1. The converse can also be proved: given a primitive arithmetic progression triple (u,v,w), 1 <= u < v < w, gcd(u,v,w) = 1, the corresponding primitive Pythagorean triple with even y is ((w-u)/2,(w+u)/2,v) or ((w+u)/2,(w-u)/2,v), depending on whether (w+u)/2 is even or odd, respectively. - Wolfdieter Lang, May 22 2013

n appears A330174(n) times. - Ray Chandler, Feb 26 2020

			Primitive Pythagorean triangle connection: a(1) = 7 because (u,v,w) = (1,5,7) corresponds to the primitive Pythagorean triangle (x = (w-u)/2, y = (w+u)/2, z = v) = (3,4,5) with leg sum 3 + 4 = 7. - _Wolfdieter Lang_, May 23 2013

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. A225949 (triangle version of leg sums).
Cf. A198384, A198385, A198386.
Cf. A058529, A198390, A198409, A198437, A330174.

a198441 n = a198441_list !! (n-1)
a198441_list = map a198390 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, 3]] (* Jean-François Alcover, Oct 22 2021 *)

A198437(n) = a(n)^2; a(n) = A198390(A198409(n)).

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).

A198436 Second 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

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

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. A198385, A198409, A198435, A198437, A198438, A198440.

a198436 n = a198436_list !! (n-1)
a198436_list = map a198385 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, 2]] (* Jean-François Alcover, Oct 20 2021 *)

a(n) = A198440(n)^2 = A198385(A198409(n)).

a(n) - A198435(n) = A198437(n) - a(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).

A144407 A058529(n+1)^2.

Original entry on oeis.org

49, 289, 529, 961, 1681, 2209, 2401, 5041, 5329, 6241, 7921, 9409, 10609, 12769, 14161, 16129, 18769, 22801, 25921, 27889, 36481, 37249, 39601, 47089, 49729, 54289, 57121, 58081, 66049, 69169, 73441, 78961, 82369, 83521, 96721, 97969

Offset: 1

Paul Curtz, Sep 30 2008

nonn

The last digit is either 1 or 9 .

Ray Chandler, Table of n, a(n) for n = 1..10000
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44.

Cf. A058529, A198437.

Select[Range[500], Union[Abs[Mod[FactorInteger[#][[All, 1]], 8, -1]]] == {1} &]^2 // Rest (* Jean-François Alcover, Sep 21 2015, after T. D. Noe *)

cp's OEIS Frontend

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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A198386 Third of a triple of squares in arithmetic progression.

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A198441 Square root of third 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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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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A198436 Second 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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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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Haskell

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A144407 A058529(n+1)^2.

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