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

1,1

Links

Table of n, a(n) for n=1..55.

Example

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, ...

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

Crossrefs

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

Sequence in context: A344991 A063185 A208854 * A201874 A059528 A070079

Adjacent sequences: A165257 A165258 A165259 * A165261 A165262 A165263

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Sep 11 2009

Status

approved