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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.

%I A364168 #95 Jun 02 2025 14:45:32
%S A364168 15,27,32,35,36,39,51,55,60,63,64,75,84,87,91,95,96,99,100,108,111,
%T A364168 115,119,123,128,132,135,140,143,144,147,155,156,159,160,171,175,180,
%U A364168 183,187,192,195,196,203,204,207,215,219,220,224,228,231,235,240,243,247,252,255
%N 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.
%C A364168 From _Darío Clavijo_, Mar 05 2025: (Start)
%C A364168 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.
%C A364168 a(n) == 0 or 3 (mod 4). (End).
%C A364168 From _Darío Clavijo_, Apr 22 2025: (Start)
%C A364168 Numbers that can be written in more than one way in the form (j+k) * (3k-j).
%C A364168 Every term is congruent to {0, 3, 4, 7, 11, 12, 15} (mod 16). (End).
%H A364168 Project Euler, <a href="https://projecteuler.net/problem=135">Problem 135: Same Differences</a>.
%e A364168 27 is a term since (6+2*3)^2 - (6+3)^2 - 6^2 = (20+2*7)^2 - (20+7)^2 - 20^2 = 27.
%o A364168 (Python)
%o A364168 from math import isqrt
%o A364168 def isok(h):
%o A364168     if (h & 15) not in [0, 3, 4, 7, 11, 12, 15]: return False
%o A364168     c = 0
%o A364168     for p in range(1, isqrt(h)+1):
%o A364168         q, r = divmod(h,p)
%o A364168         if r == 0 and (pq := p + q) & 3 == 0:
%o A364168             t = pq >> 2;
%o A364168             c += (t < p) + (p != q and t < q)
%o A364168             if c > 1: return True
%o A364168 print([h for h in range(1, 256) if isok(h)])
%Y A364168 Cf. A000005, A000290, A027750, A014601.
%K A364168 nonn
%O A364168 1,1
%A A364168 _Darío Clavijo_, Jul 12 2023