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A165260 Short legs of Primitive Pythagorean Triples which have a perimeter which is an average of a twin prime pair. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A290297 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

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Last modified March 25 20:10 EDT 2019. Contains 321477 sequences. (Running on oeis4.)