A165261 -id:A165261 - cp's OEIS frontend

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

Vladimir Joseph Stephan Orlovsky, Sep 11 2009

nonn

			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, ...

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

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

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

A165262 Sorted hypotenuses with no repeats of Primitive Pythagorean Triples (PPT) if sum of all 3 sides are averages of twin prime pairs.

Original entry on oeis.org

5, 13, 85, 113, 145, 197, 221, 241, 349, 457, 541, 569, 625, 821, 829, 841, 1025, 1037, 1093, 1157, 1241, 1433, 1465, 1621, 1741, 1769, 2029, 2069, 2249, 2353, 2441, 2465, 2501, 2669, 2725, 2801, 2809, 2825, 2873, 3029, 3077, 3221, 3293, 3305, 3389, 3889

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 11 2009

			Triples begin 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; ...
So with sorted hypotenuses:
  3 +  4 +  5 = 12, and 11 and 13 are twin primes;
  5 + 12 + 13 = 30, and 29 and 31 are twin primes; ...

Cf. A009004, A020882, A020883, A165158, A165159, A165160, A165236, A165237, A165238, A165260, A165261.

amax=10^5; 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,c]]]],{m,n+1,12!,2}],{n,1,q,1}]; Union@lst

cp's OEIS Frontend

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

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