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%I #8 Dec 12 2022 18:03:28
%S 13,37,277,313,613,7591,8209,12157,23053,32233,42953,44887,105649,
%T 225769,245941,258707,287671,331333,342049,346111,347443,393853,
%U 560719,721267,867253,1001089,1064431,1219849,1545127,1556623,1617727,1752607
%N List the first term of each triple of consecutive primes with the property that their sum is the square of a prime.
%H Robert Israel, <a href="/A130621/b130621.txt">Table of n, a(n) for n = 1..3000</a>
%e (37, 41, 43) is a triple of consecutive prime numbers; their sum is 121 which is a prime squared. Hence 37 is in the sequence.
%p f:= proc(n) local p,q,r;
%p q:= prevprime(floor(n/3));
%p p:= prevprime(q);
%p r:= nextprime(q);
%p if p+q+r = n then return p
%p elif p+q+r < n then
%p while p+q+r < n do
%p p:= q; q:= r; r:= nextprime(r);
%p od;
%p if p+q+r = n then return p fi
%p else
%p while p+q+r > n do
%p r:= q; q:= p; p:= prevprime(p);
%p od;
%p if p+q+r = n then return p fi;
%p fi;
%p false
%p end proc:
%p R:= NULL: count:= 0:
%p p:= 3:
%p while count < 100 do
%p p:= nextprime(p);
%p v:= f(p^2);
%p if v::integer then
%p R:= R,v; count:= count+1;
%p fi
%p od:
%p R; # _Robert Israel_, Sep 18 2022
%t a={};For[n=1,n<100000,n++,If[PrimeQ[Sqrt[Prime[n]+Prime[n+1]+Prime[n+2]]], AppendTo[a, Prime[n]]]]; a
%t Select[Partition[Prime[Range[132000]],3,1],PrimeQ[Sqrt[Total[#]]]&][[All,1]] (* _Harvey P. Dale_, Dec 12 2022 *)
%K nonn
%O 1,1
%A _J. M. Bergot_, Jun 18 2007
%E Edited and extended by _Stefan Steinerberger_, Jun 23 2007
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