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A263573 Intersection of A024365 and A129912.

%I A263573 #36 Nov 17 2019 02:27:18
%S A263573 6,30,60,180,210,2310,4620,60060,510510,10810800,116396280,
%T A263573 200560490130,401120980260
%N A263573 Intersection of A024365 and A129912.
%C A263573 The two sequences involve areas of primitive Pythagorean triples and primorial products. Intersections are only considered once (no repeats). Conjecture: the sequence is infinite.
%C A263573 Conjecture: The next two entries are a(12) = 200560490130, a(13) = 401120980260.
%C A263573 From _G. C. Greubel_, Dec 29 2015: (Start)
%C A263573 6|a(n) for n>=1,
%C A263573 30|a(n) for n>=2,
%C A263573 a(n)/6 = {1, 5, 10, 30, 35, 385, 770, 10010, ...} is a subset of values found in A008706.
%C A263573 (End)
%C A263573 a(12) and a(13) confirmed. a(14) > 2*10^31, if it exists. - _Giovanni Resta_, Mar 31 2017
%e A263573 A024365 begins {6, 30, 60, 84, 180, 210, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, ...}.
%e A263573 A129912 begins {1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, ...}.
%e A263573 So, common entries encountered are {6, 30, 60, 180, 210, 2310, ...}.
%e A263573 Specifically, we see that A024365(1) = A129912(3), A024365(2) = A129912(5), A024365(3) = A129912(6), A024365(5) = A129912(7).
%e A263573 These are then the first four entries of the sequence (6, 30, 60, 180).
%t A263573 s = 6 Take[Sort[(Times @@ #)/12 & /@ ({Times @@ #, (Last[#]^2 - First[#]^2)/2} & /@ Select[Subsets[Range[1, 3600, 2], {2}], GCD @@ # == 1 &])], 1800]; f[m_] := f[m] = Union[Times @@@ Subsets[FoldList[Times, 1, Prime[Range[m]]]]][[1 ;; 100]]; f[10]; f[m = 11]; While[f[m] != f[m - 1], m++]; t = f[m]; Intersection[s, t] (* _Michael De Vlieger_, Oct 22 2015, after _Harvey P. Dale_ at A020885 and _Jean-François Alcover_ at A129912 *) (* or *)
%t A263573 ok[n_] := Block[{a, f = Power @@@ FactorInteger[2 n]}, SelectFirst[ Subsets[f, {1, Floor[ Length[f]/2]}], (a = Times @@ #; IntegerQ@ Sqrt[a^2 + (2 n/a)^2]) &, {}] != {}]; pr[n_] := Product[ Prime[n+1-i]^i, {i, n}];  upto[mx_] := Block[{ric, j=1}, ric[n_, ip_, ex_] := If[n < mx, Block[{p = Prime[ip + 1]}, If[ex == 1 && ok[n], Sow@ n]; ric[n p^ex, ip + 1, ex]; If[ex > 1, ric[n p^(ex - 1), ip+1, ex-1]]]]; Sort@ Reap[ While[pr[j] < mx, ric[2^j, 1, j]; j++]][[2, 1]]]; upto[10^12] (* much faster, _Giovanni Resta_, Mar 31 2017 *)
%o A263573 (PARI)
%o A263573 \\note: code does not generate the sequence, just checks for a matching PPT entry
%o A263573 genit(area)={myMax=floor(sqrt(2*area));i5=myMax;endless=0;soln=List();
%o A263573 while(i5>=2,dun=0;j=2.*myVal/i5; k=floor(j); if(j>k, dun=1 );if(dun<1,
%o A263573 c=sqrt(i5^2 + k^2);w=floor(c);if(c>w,dun=1); if(dun<1,if(gcd(k,i5)>1,dun=1 ));
%o A263573 if(dun<1,listput(soln,k); listput(soln,i5);listput(soln,w);listsort(soln);
%o A263573 print("soln a,b,c = ", soln[1],"  ",soln[2],"  ",soln[3] );dun=2;break ));
%o A263573 i5--;endless++);if(i5<=2&&dun<1,print("no solution ") );if(i5>2&&dun<2,
%o A263573 print("max iteration limit was hit ",endless) );print (endless);}
%o A263573 (C++)
%o A263573 #include <iostream>
%o A263573 #include <fstream>
%o A263573 using namespace std;
%o A263573 int main(){ifstream fin1,fin2;
%o A263573 int myValue,myValue2,ptr,fptr,i5,j5;
%o A263573 unsigned long list1[9999]={0};
%o A263573 unsigned long list2[999]={0};
%o A263573 unsigned long final[31]={0};
%o A263573 fin1.open("A024365.txt"); fin2.open("A129912.txt");
%o A263573 ptr=1;
%o A263573 while(ptr<9999)
%o A263573 {fin1>> myValue;fin1.get();list1[ptr]=myValue;
%o A263573     if(ptr<999)
%o A263573        {fin2>> myValue2;fin2.get();list2[ptr]=myValue2;}
%o A263573     ptr++;}
%o A263573 fin1.close();fin2.close();fptr=1;
%o A263573 for(i5=1;i5<9990;i5++)
%o A263573 {for(j5=1;j5<999;j5++){
%o A263573 if(list1[i5]==list2[j5] )
%o A263573 {
%o A263573     fptr++;
%o A263573     if(fptr>30){break;}
%o A263573     final[fptr]=list1[i5];
%o A263573     cout << final[fptr] << ",";
%o A263573     break;
%o A263573 }}if(fptr>30){break;}}}
%Y A263573 Cf. A024365, A129912.
%K A263573 nonn,more
%O A263573 1,1
%A A263573 _Bill McEachen_, Oct 21 2015
%E A263573 a(12)-a(13) from _Giovanni Resta_, Mar 31 2017