A088978 Number of Pythagorean triangles having the n-th prime prime(n) as one of their sides.

0, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Author

Lekraj Beedassy, Oct 31 2003

Keywords

• nonn

Comments

Primitive Pythagorean triples are given parametrically by (M^2 - N^2)^2 + (2MN)^2 = (M^2 + N^2)^2. Odd primes are uniquely representable (ignoring signs) as M^2 - N^2, but only primes of the form 4k + 1 are uniquely representable as M^2 + N^2. Since 2MN is composite for MN > 1, an odd prime can be a side of one or two Pythagorean triangles. Thus, except for a(1) = 0, a(n) is 2 for prime(n) of the form 4k + 1, and 1 otherwise. - Chris Boyd, Jan 25 2016

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046081.

Programs

Magma

[0] cat [(4-NthPrime(n) mod 4+1)/2: n in [2..100]]; // Vincenzo Librandi, Jan 26 2016

Maple

0, seq((4-ithprime(i) mod 4 + 1)/2, i=2..1000); # Robert Israel, Jan 25 2016

Mathematica

Table[(4 - Mod[Prime@ n, 4] + 1)/2, {n, 105}] /. Rational -> 0 (* _Michael De Vlieger, Jan 26 2016 *)

PARI

a088978(n) = my(p=prime(n)); if(p==2,0,if((p-1)%4==0,2,1))
for(i=1,105,print1(a088978(i),", ")) \\ Chris Boyd, Jan 25 2016

Extensions

Corrected and extended by Ray Chandler, Nov 01 2003