A245648 The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c, where the triples consist of a triangular number, a square number and a pentagonal number.

5, 15, 145, 2775

Offset

1,1

Comments

Next term comes from a triple with c > 10^5.

From Michel Marcus, Apr 08 2021: (Start)

The 4 known triples that satisfy the requisite are [3,4,5], [9,12,15], [100, 105, 145], [900, 2625, 2775].

Let po(n) be A176774(n), the least polygonality of a number.

po([3,4,5]) = [3,4,5]; <-----

po([9,12,15]) = [4,5,3];

po([100,105,145]) = [4,3,5]; <-----

po([900,2625,2775]) = [4,5,3].

So for the 2 highlighted triples, we have a-gonal^2 + b-gonal^2 = c-gonal^2. Are there other Pythagorean triples with the same property?

Let nb(n) be A177025(n) is the number of ways to represent n as a polygonal number.

nb([3,4,5]) = [1,1,1]; <-----

nb([9,12,15]) = [4,5,3];

nb([100,105,145]) = [4,3,5];

nb([900,2625,2775]) = [4,5,3].

So for the highlighted triple, we get [1,1,1]. Are there other Pythagorean triples with the same property? (End)

Regarding the first question by Michel Marcus, if such triple [x,y,z] exists, then z > 10^4. Regarding his second question, if such triple exists, then z > 10^7. - Ivan N. Ianakiev, Dec 16 2021

a(5) > 10^11, if it exists. - Giovanni Resta, Apr 15 2021

Links

Table of n, a(n) for n=1..4.

Example

a(1) = 5 as the first such Pythagorean triple is (3,4,5). The next three triples are (9,12,15), (100,105,145), (900,2625,2775).

Mathematica

n=10^3; ppt={}; list={}; pos=1; t[x_]:=(IntegerPart[Sqrt[2*x]])*(IntegerPart[Sqrt[2*x]]+1)/2; ls[x_]:=Length[Sqrt[x]]; lis[x_]:=Length[IntegerPart[Sqrt[x]]]; lp[x_]:=Length[(Sqrt[24*x+1]+1)/6]; lip[x_]:=Length[IntegerPart[(Sqrt[24*x+1]+1)/6]]; Do[y=x+1; z=y+1; While[z<=n, While[z^2<x^2+y^2, z=z+1]; If[z^2==x^2+y^2, AppendTo[ppt, {x, y, z}]]; y=y+1], {x, 1, n}]; While[pos<Length[ppt]+1, a=ppt[[pos, 1]]; b=ppt[[pos, 2]]; c=ppt[[pos, 3]]; If[Or[And[t[a]==a, ls[b]==lis[b], lp[c]==lip[c]], And[t[a]==a, ls[c]==lis[c], lp[b]==lip[b]], And[t[b]==b, ls[a]==lis[a], lp[c]==lip[c]], And[t[b]==b, ls[c]==lis[c], lp[a]==lip[a]], And[t[c]==c, ls[a]==lis[a], lp[b]==lip[b]], And[t[c]==c, ls[b]==lis[b], lp[a]==lip[a]]], AppendTo[list, {a, b, c}]]; pos++]; l=Flatten[Sort[list, #1[[3]]<#2[[3]]&]]; Take[l, {3, -1, 3}](*Finds the terms through a search within all Pythagorean triples with c <= n*)

Crossrefs

Cf. A000217, A000290, A000326, A245646, A245647.

Cf. A176774, A177025.

Sequence in context: A124209 A207970 A207971 * A048347 A034980 A174850

Adjacent sequences: A245645 A245646 A245647 * A245649 A245650 A245651

Keyword

nonn,more,hard

Author

Ivan N. Ianakiev, Jul 28 2014

Status

approved