A364168 Numbers that can be written in more than one way in the form (j+2k)^2-(j+k)^2-j^2 with j,k>0.

15, 27, 32, 35, 36, 39, 51, 55, 60, 63, 64, 75, 84, 87, 91, 95, 96, 99, 100, 108, 111, 115, 119, 123, 128, 132, 135, 140, 143, 144, 147, 155, 156, 159, 160, 171, 175, 180, 183, 187, 192, 195, 196, 203, 204, 207, 215, 219, 220, 224, 228, 231, 235, 240, 243, 247, 252, 255

Offset: 1

Author

Darío Clavijo, Jul 12 2023

Keywords

• nonn

Comments

From Darío Clavijo, Mar 05 2025: (Start)

Also, numbers h that can be written as a difference of squares such as h=4*y^2-x^2 where x=2*y-p and y=(p+q)/4 and p<3*q with p and q divisors of h.

a(n) == 0 or 3 (mod 4). (End).

From Darío Clavijo, Apr 22 2025: (Start)

Numbers that can be written in more than one way in the form (j+k) * (3k-j).

Every term is congruent to {0, 3, 4, 7, 11, 12, 15} (mod 16). (End).

Examples

			27 is a term since (6+2*3)^2 - (6+3)^2 - 6^2 = (20+2*7)^2 - (20+7)^2 - 20^2 = 27.
		

Links

• Project Euler, Problem 135: Same Differences.

Crossrefs

Cf. A000005, A000290, A027750, A014601.

Programs

Python

from math import isqrt
def isok(h):
    if (h & 15) not in [0, 3, 4, 7, 11, 12, 15]: return False
    c = 0
    for p in range(1, isqrt(h)+1):
        q, r = divmod(h,p)
        if r == 0 and (pq := p + q) & 3 == 0:
            t = pq >> 2;
            c += (t < p) + (p != q and t < q)
            if c > 1: return True
print([h for h in range(1, 256) if isok(h)])