A376208 Numbers k such that 4*k+1 is the hypotenuse of a primitive Pythagorean triangle with an even short leg.

4, 7, 9, 13, 16, 18, 21, 25, 27, 31, 34, 36, 43, 46, 49, 51, 55, 57, 60, 64, 66, 70, 73, 76, 81, 87, 91, 93, 94, 99, 100, 102, 111, 112, 114, 121, 123, 126, 127, 133, 136, 141, 144, 148, 150, 156, 157, 160, 165, 169, 171, 172, 175, 181, 183, 186, 189, 196, 198, 202

Offset: 1

Author

Hugo Pfoertner, Sep 20 2024

Keywords

• nonn

Comments

This sequence is not A094178 \ A375750, because there are hypotenuses for which both kinds of triangles exist. The smallest example occurs for hypotenuse c = 4*a(5) + 1 = 65. The triangle (16, 63, 65) has an even short leg, but there is also the triangle (33, 56, 65) with an odd short leg. Thus, 16 = (65-1)/4 is a term in this sequence and in A375750.

Sorted distinct values of ({A081985} - 1)/4.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Crossrefs

{a(n)} U A375750 = A094178.

Cf. A004613, A008846, A081985.

Programs

PARI

is_a376208(n,r=0) = my(c=4*n+1, q=qfbsolve(Qfb(1,0,1), c^2, 3), qd=#q, is=0); for(k=1, qd-1, if(vecmin(abs(q[k]))%2==r && gcd([c,q[k]])==1, is=1; break)); is

Python

# for an array from the beginning
from math import gcd, isqrt
test_all_k_upto = 202
A376208, limit = set(), test_all_k_upto * 4 + 1
for x in range(2,isqrt(limit)+1):
    for y in range(min(((d:=isqrt(2*x**2)-x))-(d%2==x%2), (yy:=isqrt(limit-x**2))-(yy%2==x%2)),0,-2):
        if gcd(x, y) == 1: A376208.add((x**2 + y**2 - 1) // 4)
print(A376208:=sorted(A376208)) # Karl-Heinz Hofmann, Sep 28 2024

Python

# for testing high single terms
from math import isqrt, gcd
from sympy import factorint
def A376208_isok(k):
    c  = k * 4 + 1
    if any([(pf-1) % 4 for pf in factorint(c)]): return False # (Test imported from A008846)
    y2 = c - (x2:=(x:=isqrt(c))**2)
    while 2*x*(y:=isqrt(y2)) < x2-y2:
        if y2 == y**2 and gcd(x, y) == 1: return True
        x -= 1
        y2 = c - (x2:=x**2) # Karl-Heinz Hofmann, Oct 17 2024