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%I A245646 #20 Dec 23 2023 10:56:51
%S A245646 3,9,100,900
%N A245646 The smallest member 'a' 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.
%C A245646 The next term (if it exists) comes from a triple with c > 10^5.
%e A245646 a(1) = 3 as the first such Pythagorean triple is (3,4,5). The next three triples are (9,12,15), (100,105,145), (900,2625,2775).
%t A245646 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,{1,-1,3}](*Finds the terms through a search within all Pythagorean triples with c <= n*)
%Y A245646 Cf. A000217, A000290, A000326, A245647, A245648
%K A245646 nonn,more,hard
%O A245646 1,1
%A A245646 _Ivan N. Ianakiev_, Jul 28 2014