A307749 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.

13, 53, 97, 137, 233, 313, 421, 461, 641, 821, 877, 929, 997, 1061, 1093, 1129, 1201, 1217, 1229, 1693, 1709, 1873, 2213, 2309, 3001, 3049, 3169, 3181, 3469, 3517, 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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..55.

Example

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.

Prog

(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; }
isok(p) = isprime(p) && (q=is(p)) && is(q);
lista(nn) = T=thueinit('x^2+1, 1); forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, May 01 2019

Crossrefs

Cf. A002144, A283391, A307718.

Sequence in context: A262447 A165352 A262287 * A031905 A214523 A087880

Adjacent sequences: A307746 A307747 A307748 * A307750 A307751 A307752

Keyword

nonn

Author

Torlach Rush, Apr 26 2019

Status

approved