A307749 - OEIS

%I #30 May 01 2019 09:14:01

%S 13,53,97,137,233,313,421,461,641,821,877,929,997,1061,1093,1129,1201,

%T 1217,1229,1693,1709,1873,2213,2309,3001,3049,3169,3181,3469,3517,

%U 3581,3593,3677,3701,3733,3881,3917,4057,4397,4409,4621,4813,5237,5437,5441,5953,6257,6301,6577,6637,6661,6857,7229,7481,7669

%N Lengths of the hypotenuse of primitive pythagorean triples if prime, whose shorter legs sum to the hypotenuse of prime length of another primitive pythagorean triple whose shorter legs sum to a prime number.

%C Embedded in this sequence are subsets based on the definition, for example {97,137}, and {3049,3881,5441,7481}. These arise when terms are both the length of the hypotenuse of one primitive Pythagorean triple and the sum of the two shorter legs of another.

%e 13 is a term because 13^2 = 12^2 + 5^2 and 12 + 5 = 17 and 17^2 = 15^2 + 8^3 and 15 + 8 = 23.

%o (PARI) is(n) = {if((n%4 != 1) || !isprime(n), return(0)); my(v=thue(T, n^2), q); for(i=1, #v, if(v[i][1]>0 && v[i][2]>=v[i][1] && (q=vecsum(v[i])) && isprime(q), return(q)); ); 0;}

%o isok(p) = isprime(p) && (q=is(p)) && is(q);

%o lista(nn) = T=thueinit('x^2+1, 1); forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ _Michel Marcus_, May 01 2019

%Y Cf. A002144, A283391, A307718.

%K nonn

%O 1,1

%A _Torlach Rush_, Apr 26 2019