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A263573 Intersection of A024365 and A129912. 0
6, 30, 60, 180, 210, 2310, 4620, 60060, 510510, 10810800, 116396280, 200560490130, 401120980260 (list; graph; refs; listen; history; text; internal format)
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 two next 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, 15, 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-Fran├žois 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: A116360 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

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Last modified April 22 21:39 EDT 2018. Contains 302916 sequences. (Running on oeis4.)