A307718 Primes p such that a + b is prime and a^2 + b^2 = p^2 and p = c + d such that c^2 + d^2 = e^2.

17, 73, 97, 113, 137, 193, 233, 313, 337, 401, 449, 457, 521, 569, 641, 673, 809, 929, 977, 1009, 1049, 1129, 1153, 1201, 1217, 1249, 1289, 1297, 1361, 1409, 1481, 1609, 1697, 1873, 1889, 1913, 2017, 2137, 2153, 2273, 2281, 2377, 2393, 2417, 2441, 2521, 2969, 3001

Offset

1,1

Comments

a^2 + b^2 = p^2 is a primitive Pythagorean triple since the hypotenuse is prime.

c^2 + d^2 = e^2 also appears to be a primitive Pythagorean triple.

Links

Table of n, a(n) for n=1..48.

Example

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

Prog

(PARI) isok2(p) = {for (k=1, p-1, if (issquare(zz=k^2+(p-k)^2), return (zz); ); ); return(0); }
isok1(p) = {forprime (pp=2, 2*p, for (i=1, pp-1, if (issquare(z=i^2+(pp-i)^2) && (p==sqrtint(z)) && (zz=isok2(p)), return (1); ); ); ); return(0); }
isok(p) = isprime(p) && isok1(p); \\ Michel Marcus, Apr 26 2019
(PARI) \\ uses isok2 from above but much quicker version
is(n)=if(n%4 != 1 || !isprime(n), return(0)); my(v=thue(T, n^2)); for(i=1, #v, if(v[i][1]>0 && v[i][2]>=v[i][1] && isprime(vecsum(v[i])), return(1))); 0; \\ A283391
lista(nn) = T=thueinit('x^2+1, 1); forprime(p=2, nn, if (is(p) && isok2(p), print1(p, ", "))); \\ Michel Marcus, Apr 27 2019

Crossrefs

Subsequence of A002144. Subsequence of A283391.

Sequence in context: A039371 A245042 A144245 * A139919 A043194 A043974

Adjacent sequences: A307715 A307716 A307717 * A307719 A307720 A307721

Keyword

nonn

Author

Torlach Rush, Apr 23 2019

Status

approved