%I #6 Feb 23 2019 19:36:22

%S 17,31,41,71,89,97,109,113,127,197,239,251,349,379,433,461,631,829,

%T 881,911,919,953,967,991,1151,1231,1427,1429,1471,1693,1759,1847,1871,

%U 2143,2269,2273,2393,2399,2437,2531,2591,2617,2633,2647,2729,2851,2953,2969

%N Primes of the form p + q + r with p, q and r also prime such that p^2 + 1 = q^2 + r^2.

%e Primes p=13, q=7, and r=11 give 13^2 + 1 = 7^2 + 11^2 = 170, so 13 + 7 + 11 = 31 (prime) is a term.

%e Primes p=17, q=11, and r=13 give 17^2 + 1 = 11^2 + 13^2 = 290, so 17 + 11 + 13 = 41 (prime) is a term.

%e Primes p=23, q=13, and r=19 give 23^2 + 1 = 13^2 + 19^2 = 530, but 23 + 13 + 19 = 55 (composite), so 55 is not a term.

%e Primes p=31, q=11, and r=29 give 31^2 + 1 = 11^2 + 29^2 = 962, so 31 + 11 + 29 = 71 (prime) is a term.

%p From _Emeric Deutsch_, Jun 21 2009: (Start)

%p The second Maple program yields the pairs [x+y+z,[x,y,z]].

%p A := {}: for i to 250 do for j to 250 do for k to 250 do x := ithprime(i): y := ithprime(j): z := ithprime(k): if `and`(isprime(x+y+z) = true, x^2+1 = y^2+z^2) then A := `union`(A, {x+y+z}) else end if end do end do end do: A; # end of the program

%p B := {}: for i to 20 do for j to 20 do for k to 20 do x := ithprime(i): y := ithprime(j): z := ithprime(k): if `and`(isprime(x+y+z) = true, x^2+1 = y^2+z^2) then B := `union`(B, {[x+y+z, [x, y, z]]}) else end if end do end do end do: B; # end of the program

%p (End)

%Y Cf. A000040.

%K nonn

%O 1,1

%A Vladislav-Stepan Malakhovsky and _Juri-Stepan Gerasimov_, May 31 2009

%E Corrected and extended by _Emeric Deutsch_, Jun 21 2009

%E Edited by _Jon E. Schoenfield_, Feb 23 2019