A263573 Intersection of A024365 and A129912.

6, 30, 60, 180, 210, 2310, 4620, 60060, 510510, 10810800, 116396280, 200560490130, 401120980260

Offset

1,1

Comments

The two sequences involve areas of primitive Pythagorean triples and primorial products. Intersections are only considered once (no repeats). Conjecture: the sequence is infinite.

Conjecture: The next two entries are a(12) = 200560490130, a(13) = 401120980260.

From G. C. Greubel, Dec 29 2015: (Start)

6|a(n) for n>=1,

30|a(n) for n>=2,

a(n)/6 = {1, 5, 10, 30, 35, 385, 770, 10010, ...} is a subset of values found in A008706.

(End)

a(12) and a(13) confirmed. a(14) > 2*10^31, if it exists. - Giovanni Resta, Mar 31 2017

Links

Table of n, a(n) for n=1..13.

Example

A024365 begins {6, 30, 60, 84, 180, 210, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, ...}.
A129912 begins {1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, ...}.
So, common entries encountered are {6, 30, 60, 180, 210, 2310, ...}.
Specifically, we see that A024365(1) = A129912(3), A024365(2) = A129912(5), A024365(3) = A129912(6), A024365(5) = A129912(7).
These are then the first four entries of the sequence (6, 30, 60, 180).

Mathematica

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 *)
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 *)

Prog

(PARI)
\\note: code does not generate the sequence, just checks for a matching PPT entry
genit(area)={myMax=floor(sqrt(2*area)); i5=myMax; endless=0; soln=List();
while(i5>=2, dun=0; j=2.*myVal/i5; k=floor(j); if(j>k, dun=1 ); if(dun<1,
c=sqrt(i5^2 + k^2); w=floor(c); if(c>w, dun=1); if(dun<1, if(gcd(k, i5)>1, dun=1 ));
if(dun<1, listput(soln, k); listput(soln, i5); listput(soln, w); listsort(soln);
print("soln a, b, c = ", soln[1], "  ", soln[2], "  ", soln[3] ); dun=2; break ));
i5--; endless++); if(i5<=2&&dun<1, print("no solution ") ); if(i5>2&&dun<2,
print("max iteration limit was hit ", endless) ); print (endless); }
(C++)
#include <iostream>
#include <fstream>
using namespace std;
int main(){ifstream fin1, fin2;
int myValue, myValue2, ptr, fptr, i5, j5;
unsigned long list1[9999]={0};
unsigned long list2[999]={0};
unsigned long final[31]={0};
fin1.open("A024365.txt"); fin2.open("A129912.txt");
ptr=1;
while(ptr<9999)
{fin1>> myValue; fin1.get(); list1[ptr]=myValue;
    if(ptr<999)
       {fin2>> myValue2; fin2.get(); list2[ptr]=myValue2; }
    ptr++; }
fin1.close(); fin2.close(); fptr=1;
for(i5=1; i5<9990; i5++)
{for(j5=1; j5<999; j5++){
if(list1[i5]==list2[j5] )
{
    fptr++;
    if(fptr>30){break; }
    final[fptr]=list1[i5];
    cout << final[fptr] << ", ";
    break;
}}if(fptr>30){break; }}}

Crossrefs

Cf. A024365, A129912.

Sequence in context: A336219 A065800 A181827 * A145010 A056835 A056836

Adjacent sequences: A263570 A263571 A263572 * A263574 A263575 A263576

Keyword

nonn,more

Author

Bill McEachen, Oct 21 2015

Extensions

a(12)-a(13) from Giovanni Resta, Mar 31 2017

Status

approved