%I
%S 240,599,1197,1957,2777,4475,6870,9727,13111,16006,19318,24588,30446,
%T 37372,43923,52863,59912,68278,79653,93050,109121,125459,138200,
%U 146888,156205,175051,201823,236438,255780,282105,307211,338310,365530,397086
%N Floor of the area of consecutive PrimeIndexed Prime triangles.
%C Conjecture: The triples (3,5,11), (5,11,17), (11,17,31) are the only consecutive PIP triples that cannot form a triangle.
%F A prime index is the numerical position of a prime number in the sequence of prime numbers. A PrimeIndexed Prime (PIP) is a prime number whose index is also prime. A PrimeIndexed Prime triangle is a triangle whose sides are PrimeIndexed Primes.
%e The first set of consecutive PIPs that produces a triangle, 17, 31 and 41, produces the 17 X 31 X 41 unit triangle whose area is 240.462... square units.
%t ar[n_]:=Module[{a=First[n],b=n[[2]],c=Last[n],s},s=Total[{a,b,c}]/2;Sqrt[s(sa)(sb)(sc)]]; Floor[ar[#]]&/@Partition[Select[Prime[ Range[6,200]], PrimeQ[PrimePi[#]]&],3,1] (* _Harvey P. Dale_, Jun 29 2011 *)
%o (PARI) area(n) = for(x=1,n,a=prime(prime(x));b=prime(prime(x+1));c=prime(prime(x+2)); if(a+b<=c,p=a+b+c;y =1/4*sqrt(p*(p2*a)*(p2*b)*(p2*c));print1(floor(y)",")))
%K nonn
%O 1,1
%A _Cino Hilliard_, Jul 28 2006
