A130621 List the first term of each triple of consecutive primes with the property that their sum is the square of a prime.

13, 37, 277, 313, 613, 7591, 8209, 12157, 23053, 32233, 42953, 44887, 105649, 225769, 245941, 258707, 287671, 331333, 342049, 346111, 347443, 393853, 560719, 721267, 867253, 1001089, 1064431, 1219849, 1545127, 1556623, 1617727, 1752607

Offset: 1

Author

J. M. Bergot, Jun 18 2007

Keywords

• nonn

Examples

			(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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3000

Programs

Maple

f:= proc(n) local p,q,r;
   q:= prevprime(floor(n/3));
   p:= prevprime(q);
   r:= nextprime(q);
   if p+q+r = n then return p
   elif p+q+r < n then
     while p+q+r < n do
       p:= q; q:= r; r:= nextprime(r);
     od;
     if p+q+r = n then return p fi
   else
     while p+q+r > n do
       r:= q; q:= p; p:= prevprime(p);
     od;
     if p+q+r = n then return p fi;
   fi;
   false
end proc:
R:= NULL: count:= 0:
p:= 3:
while count < 100 do
  p:= nextprime(p);
  v:= f(p^2);
  if v::integer then
    R:= R,v; count:= count+1;
  fi
od:
R; # Robert Israel, Sep 18 2022

Mathematica

a={};For[n=1,n<100000,n++,If[PrimeQ[Sqrt[Prime[n]+Prime[n+1]+Prime[n+2]]], AppendTo[a, Prime[n]]]]; a
Select[Partition[Prime[Range[132000]],3,1],PrimeQ[Sqrt[Total[#]]]&][[All,1]] (* Harvey P. Dale, Dec 12 2022 *)

Extensions

Edited and extended by Stefan Steinerberger, Jun 23 2007