This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A259060 #31 Mar 18 2025 21:07:23
%S A259060 6426,7900,9614,11592,13858,16436,19350,22624,26282,30348,34846,39800,
%T A259060 45234,51172,57638,64656,72250,80444,89262,98728,108866,119700,131254,
%U A259060 143552,156618,170476,185150,200664,217042,234308,252486,271600,291674
%N A259060 Numbers that are representable in at least two ways as sums of four distinct nonvanishing cubes.
%C A259060 This is the second part of Exercise 229 in Sierpiński's problem book. See p. 20, and p. 110 for the solution. He uses the identity (n-8)^3 + (n-1)^3 + (n+1)^3 + (n+8)^3 = 4*n^3 + 390 = (n-7)^3 + (n-4)^3 + (n+4)^3 + (n+7)^3, for n >= 9.
%C A259060 Here n is replaced by n + 9: (n+1)^3 + (n+8)^3 + (n+10)^3 + (n+17)^3 = 4*n^3 + 108*n^2 + 1362*n + 6426 = (n+2)^3 + (n+5)^3 + (n+13)^3 + (n+16)^3, for n >= 0.
%C A259060 There may be other numbers with this properties.
%C A259060 Because the summands have no common factor > 1 each of these two representations is called primitive. - _Wolfdieter Lang_, Aug 20 2015
%D A259060 W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970.
%H A259060 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A259060 a(n) = (2*(n+9))*(2*n^2+36*n+357) = 2*A261241(n), n >= 0. See the comment for the sum of four distinct cubes in two different ways.
%F A259060 O.g.f.: 2*(3213 - 8902*x + 8285*x^2 - 2584*x^3) / (1-x)^4.
%F A259060 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Vincenzo Librandi_, Aug 13 2015
%e A259060 a(0) = 6426 = 1^3 + 8^3 + 10^3 + 17^3 = 2^3 + 5^3 + 13^3 + 16^3.
%e A259060 a(1) = 7900 = 2^3 + 9^3 + 11^3 + 18^3 = 3^3 + 6^3 + 14^3 + 17^3.
%t A259060 CoefficientList[Series[2 (3213 - 8902 x + 8285 x^2 - 2584 x^3)/(1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 13 2015 *)
%t A259060 LinearRecurrence[{4,-6,4,-1},{6426,7900,9614,11592},40] (* _Harvey P. Dale_, Sep 30 2016 *)
%o A259060 (Magma) [(2*(n+9))*(2*n^2+36*n+357): n in [0..50]]; // _Vincenzo Librandi_, Aug 13 2015
%o A259060 (Magma) I:=[6426,7900,9614,11592]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Aug 13 2015
%Y A259060 Cf. A261241, A259058 (squares).
%K A259060 nonn,easy
%O A259060 0,1
%A A259060 _Wolfdieter Lang_, Aug 12 2015