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

1,3

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

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 *)

Prog

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

Crossrefs

Cf. A046081.

Sequence in context: A264840 A308188 A046219 * A276948 A160245 A154351

Adjacent sequences: A088975 A088976 A088977 * A088979 A088980 A088981

Keyword

nonn

Author

Lekraj Beedassy, Oct 31 2003

Extensions

Corrected and extended by Ray Chandler, Nov 01 2003

Status

approved