A165260 Short legs of primitive Pythagorean triples which have a perimeter which is the average of a twin prime pair.

3, 5, 15, 21, 24, 28, 36, 41, 59, 64, 89, 100, 101, 120, 131, 132, 141, 153, 155, 168, 180, 203, 204, 208, 209, 215, 220, 231, 244, 280, 288, 300, 309, 315, 336, 341, 348, 351, 395, 405, 408, 429, 448, 453, 455, 495, 520, 540, 551, 567, 568, 580, 592, 636, 648

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Sep 11 2009

Keywords

• nonn

Examples

			Triples (a,b,c) which satisfy the rules are (3,4,5), (5,12,13), (15,112,113), (21,220,221), (24,143,145), (28,195,197), (36,77,85), (41,840,841), (59,1740,1741), (64,1023,1025), (89,3960,3961), (100,2499,2501), ... 3+4+5=12 -> 11 and 13 are primes, 5+12+13=30 -> 29 and 31 are primes, ...
		

Crossrefs

Cf. A020884, A014574, A009004, A165261, A165262.

Programs

Maple

isA014574 := proc(n)
        return ( isprime(n-1) and isprime(n+1) ) ;
end proc:
isA165260 := proc(n)
        local d,bplc,b,c ;
        for d in numtheory[divisors](n^2) do
                bplc := n^2/d ;
                c := (d+bplc)/2 ;
                b := (bplc-d)/2 ;
                if type(c,'integer') and type(b,'integer') then
                if c > b and b >= n then
                        if igcd(n,b,c) = 1 and  isA014574(n+b+c) then
                                return true;
                        end if;
                end if;
                end if;
        end do:
        return false;
end proc:
for n from 3 to 600 do
        if isA165260(n) then
                printf("%d,",n);
        end if;
end do: # R. J. Mathar, Oct 29 2011

Mathematica

amax=10^4;lst={};k=0;q=12!;Do[If[(e=((n+1)^2-n^2))>amax,Break[]];Do[If[GCD[m,n]==1,a=m^2-n^2;b=2*m*n;If[GCD[a,b]==1,If[a>b,{a,b}={b,a}];If[a>amax,Break[]];c=m^2+n^2;x=a+b+c;If[PrimeQ[x-1]&&PrimeQ[x+1],k++;AppendTo[lst,a]]]],{m,n+1,12!,2}],{n,1,q,1}];Union@lst