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
Keywords
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, ...
Programs
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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
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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