A190797 For primes p and q=p+6 create primitive Pythagorean triangles with sides (q^2 - p^2)/2, (p^2 + q^2)/2, and p*q. If the two remainders of the middle and longest side modulo the shortest side are both prime, then p is in the sequence.

11, 23, 41, 83, 107, 167, 191, 263, 307, 347, 367, 461, 641, 653, 877, 881, 1103, 1187, 1367, 2081, 2393, 2677, 3607, 4283, 4357, 4967, 5081, 5231, 5387, 5471, 5651, 6037, 6197, 6311, 6353, 6857, 7823, 8117, 8693, 8747, 9221, 9743, 9851, 9923

Offset

1,1

Comments

The short side is 6p+18, the middle side p^2+6p, the long side 6p+18+p^2.

The first few values have more terms == 3 (mod 4) than 1 (mod 4), but this does not appear to be the case for later terms. - Franklin T. Adams-Watters, May 22 2011

Links

Table of n, a(n) for n=1..44.

Formula

If p=6k+5, then the remainders are 7 + 12*k and 25 + 12*k.

If p=6k+1, then the remainders are 7 + 24*k and 25 + 24*k.

Example

For p=41 and q=47, the sides are (47^2 - 41^20)/2=264, 41*47=1927 and (41^2 + 43^2)/2=1945; divide 1927 and 1945 through 264 to get remainders 79 and 97.  Since both are primes, p=41 is in the sequence.

Prog

(PARI) forprime(p=5, 10000, if(isprime(q=p+6), x=(q^2-p^2)/2; if(isprime(((q^2+p^2)/2)%x)&isprime(p*q%x), print1(p", "))))

Crossrefs

Cf. A023201.

Sequence in context: A060915 A052034 A077041 * A068842 A357364 A199848

Adjacent sequences: A190794 A190795 A190796 * A190798 A190799 A190800

Keyword

nonn

Author

J. M. Bergot, May 20 2011

Extensions

Corrected and extended by Franklin T. Adams-Watters, May 22 2011

Status

approved