A374263 Number of distinct primitive Pythagorean triples (j^2 - k^2, 2*j*k, j^2 + k^2) where 1 <= k < j <= n.

1, 2, 4, 6, 8, 11, 15, 18, 22, 27, 31, 37, 43, 47, 55, 63, 69, 78, 86, 92, 102, 113, 121, 131, 143, 152, 164, 178, 186, 201, 217, 227, 243, 255, 267, 285, 303, 315, 331, 351, 363, 384, 404, 416, 438, 461, 477, 498

Offset: 2

Author

David Roggeveen Byrne, Jul 01 2024

Keywords

• nonn

Comments

Triples of this form are primitive and distinct when j,k are coprime (i.e., gcd(j,k) = 1) and of opposite parity (i.e., j+k == 0 (mod 2)).

Examples

			For n=5, the possible pairs for j,k are
              Generated  Primitive      As it's included on
               triple     triple        the list, is it new?
  j=2, k=1 ->  3, 4, 5    3, 4, 5            Yes
  j=3, k=1 ->  8, 6,10    3, 4, 5            No
  j=3, k=2 ->  5,12,13    5,12,13            Yes
  j=4, k=1 -> 15, 8,17    8,15,17            Yes
  j=4, k=2 -> 12,16,20    3, 4, 5            No
  j=4, k=3 ->  7,24,25    7,24,25            Yes
  j=5, k=1 -> 24,10,26    5,12,13            No
  j=5, k=2 -> 21,20,29   20,21,29            Yes
  j=5, k=3 -> 16,30,34    8,15,17            No
  j=5, k=4 ->  9,40,41    9,40,41            Yes
Among these there are a(5) = 6 distinct primitive triples.
		

Crossrefs

Cf. A049690, A055034.

Programs

Python

from sympy import totient
def A374263(n): return (sum(totient(n) for n in range(1,n+1,2))>>1) + sum(totient(n) for n in range(2,n+1,2)) # Chai Wah Wu, Aug 04 2024

Formula

a(n) = Sum_{i=2..n} A055034(i).

a(n) = (A049690(n) - 1)/2. - Hugo Pfoertner, Jul 16 2024