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A360427 Values of the argument at successive record minima of the function R defined as follows. For any integer x >= 1, let y > x be the smallest integer such that there exist integers x < c < d < y such that x^3 + y^3 = c^3 + d^3. Then R(x) = y/x.

%I A360427 #41 Jan 08 2025 11:38:15
%S A360427 1,2,8,9,10,17,30,42,51,135,156,285,792,1634,3751,4026,6192,14934,
%T A360427 15768,16147,45121,58230,61389,79876,167757,177560,213652,525537,
%U A360427 917324,1050787,2237052,3954983,4157802
%N A360427 Values of the argument at successive record minima of the function R defined as follows. For any integer x >= 1, let y > x be the smallest integer such that there exist integers x < c < d < y such that x^3 + y^3 = c^3 + d^3. Then R(x) = y/x.
%C A360427 For a given integer x, the identity x^3 + (12x)^3 = (9x)^3 + (10x)^3 holds, so R(x) <= 12.
%C A360427 A quadruple x = 2*N^4 - 4*N^3 + 9*N^2 - 8*N +10, y = 2*N^4 + 6*N^2 + N + 9, c = 2*N^4 - 3*N^3 + 12*N^2 - 5*N + 12, d = 2*N^4 - N^3 + 6*N^2 + N + 1 (for integer N) shows that the sequence is infinite.
%H A360427 Giedrius Alkauskas, <a href="https://web.vu.lt/mif/g.alkauskas/math/cubes.pdf">On the sequence representing narrowest intervals containing two pairs of integers with equal cube sums</a>.
%H A360427 Ajai Choudhry, <a href="https://doi.org/10.1216/rmjm/1181071714">On equal sums of cubes</a>, Rocky Mountains Journal of Mathematics, 28 (4), 1998.
%e A360427 For x = 1, y = 12, 1^3 + 12^3 = 9^3 + 10^3, R(1) = 12. So, a(1) = 1.
%e A360427 For x = 2, y = 16, 2^3 + 16^3 = 9^3 + 15^3, R(2) = 8. So, a(2) = 2.
%e A360427 For x = 3, y = 36, 3^3 + 36^3 = 27^3 + 30^3, R(3) = 12. So, this does not provide a record minimum. The same negative outcome happens for x = 4, x = 5, x = 6, x = 7.
%e A360427 For x = 8, y = 53, 8^3 + 53^3 = 29^3 + 50^3, R(8) = 6.625. So, a(4) = 8.
%e A360427 For n = 8, a(8) = 42, since 42^3 + 69^3 = 56^3 + 61^3, and the ratio R(42) = 69/42 = 1.6428571... is an absolute minimum (eighth successive) for the function R(x) for 1 <= x <= 42.
%o A360427 (Python)
%o A360427 xm,ym,x,n = 0,1,0,1
%o A360427 while True:
%o A360427     x,y = x+1,x+4
%o A360427     while y*xm < ym*x:
%o A360427         c,d,s = x+1,y-1,x**3+y**3
%o A360427         while c<d:
%o A360427             t=c**3+d**3
%o A360427             if t<s:
%o A360427                 c+=1
%o A360427             elif t>s:
%o A360427                 d-=1
%o A360427             else:
%o A360427                 break
%o A360427         if t==s:
%o A360427             print("a({})={} x={} c={} d={} y={}".format(n,x,x,c,d,y))
%o A360427             xm,ym,n = x,y,n+1
%o A360427             break
%o A360427         y+=1
%o A360427 # _Bert Dobbelaere_, Mar 18 2023
%Y A360427 Cf. A001235, A011541.
%K A360427 nonn,more
%O A360427 1,2
%A A360427 _Giedrius Alkauskas_, Feb 07 2023
%E A360427 a(25)-a(33) from _Bert Dobbelaere_, Mar 18 2023