A081985 -id:A081985 - cp's OEIS frontend

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

Hugo Pfoertner, Sep 20 2024

nonn

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.

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

{a(n)} U A375750 = A094178.

Cf. A004613, A008846, A081985.

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

# 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

# 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

A087937 Prime hypotenuse of primitive Pythagorean triangles with even short leg (or odd long leg).

Original entry on oeis.org

17, 29, 37, 53, 73, 101, 109, 137, 173, 197, 229, 241, 257, 281, 293, 349, 373, 397, 401, 409, 449, 457, 509, 577, 593, 601, 641, 661, 677, 701, 733, 757, 809, 829, 857, 877, 941, 977, 997, 1021, 1033, 1049, 1061, 1093, 1153, 1181, 1193, 1229, 1237, 1277

Offset: 1

Lekraj Beedassy, Oct 27 2003

nonn

Ordered intersection of A002144 and A081985.

These same triangles have the property that sqrt(h+c) = t, where h = a(n), c is the even short leg, t is an integer and the odd long leg is an odd multiple of t, but it appears there are no cases where that multiple is 1, in contrast with the similar property that applies to the triangles of A087937. Also see A008846. - Richard R. Forberg, May 11 2016

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A002144, A008846, A081985, A087937.

More terms from Ray Chandler, Oct 28 2003

cp's OEIS Frontend

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

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