A369498 - OEIS

%I #21 Mar 02 2024 23:19:28

%S 65,260,585,650,1037,1040,1625,1853,2340,2378,2465,2600,3185,3650,

%T 4148,4160,5265,5513,5850,6500,6890,7298,7412,7865,8177,9333,9360,

%U 9512,9593,9860,10400,10985,12740,14600,14625,14690,16133,16250,16592,16640,16677,18005

%N Integers k such that k = a^2 + b^2 = c^2 + d^2 and a + b = 3(c - d), where a, b, c and d are distinct positive integers.

%C These numbers allow the generation of infinitely many arithmetic progressions (A.P.) of length 4 whose elements belong to A000404.

%C For example, (37, 61, 85, 109) is an A.P. whose difference is 24, and 37, 61, 85 and 109 are in A000404.

%C To prove that in A000404 there exist infinitely many 4-tuples (x,y,z,w) that form an A.P. we can find a 4-tuple as a function of a parameter m. For this purpose, we consider the following expressions:

%C     x = (m - a)^2 + (m - b)^2 = 2m^2 - 2m(a + b) + a^2 + b^2

%C     y = (m - c)^2 + (m + d)^2 = 2m^2 - 2m(c - d) + c^2 + d^2

%C     z = (m + c)^2 + (m - d)^2 = 2m^2 + 2m(c - d) + c^2 + d^2

%C     w = (m + a)^2 + (m + b)^2 = 2m^2 + 2m(a + b) + a^2 + b^2

%C where a, b, c, d are distinct integers such that a^2 + b^2 = c^2 + d^2. Therefore, x, y, z, w will be in A.P. if x + z = 2y, whence we conclude that a + b = 3(c-d).

%C Thus, if k = a^2 + b^2 = c^2 + d^2 and a + b = 3(c - d), where a, b, c and d are distinct positive integers, then (x, y, z, w) form an A.P. for all positive integers m, and if m > min{a,b,c,d} then all elements belong to A000404.

%C The smallest number with this property is 65, since 65 = 8^2 + 1^2 = 7^2 + 4^2 and 8 + 1 = 3*(7 - 4). Taking k = 65 and m = 2, the tuple (37, 61, 85, 109) results.

%H D. R. Heath-Brown, <a href="https://www.researchgate.net/publication/246697804_Linear_relations_amongst_sums_of_two_squares">Linear relations amongst sums of two squares</a>, London Mathematical Society Lecture Note Series, 303 (2003), 133 - 176.

%e 1037 is a term because 1037 = 26^2 + 19^2 = 29^2 + 14^2 and 26 + 19 = 3*(29 - 14).

%o (Python)

%o from math import isqrt

%o def A369498_list(n):

%o     return sorted([

%o         a**2 + b**2

%o         for a in range(1, isqrt(n) + 1)

%o         for b in range(1, a)

%o         for c in range(1, isqrt(n) + 1)

%o         for d in range(1, c)

%o         if a != c and a != d

%o         and a**2 + b**2 == c**2 + d**2

%o         and a + b == 3 * (c - d)

%o         and a**2 + b**2 <= n

%o     ])

%o print(A369498_list(18500))

%Y Cf. A000404, A007692.

%K nonn

%O 1,1

%A _Gonzalo Martínez_, Jan 24 2024