A016078 Smallest number that is sum of 2 positive n-th powers in 2 different ways.
4, 50, 1729, 635318657
Offset: 1
Examples
4 = 1^1 + 3^1 = 2^1 + 2^1; 50 = 1^2 + 7^2 = 5^2 + 5^2, 1729 = 1^3 + 12^3 = 9^3 + 10^3; 635318657 = 59^4 + 158^4 = 133^4 + 134^4 = A018786(1).
Links
- R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.
- Eric Weisstein's World of Mathematics, Diophantine Equation--5th Powers
Programs
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Mathematica
(* This is just an empirical verification *) Do[max = 4 + n^4; Clear[cnt]; cnt[] = 0; smallest = Infinity; Do[ cnt[an = x^n + y^n] += 1; If[cnt[an] == 2 && an < smallest, smallest = an], {x, 1, max}, {y, x, max}]; Print["a(", n, ") = ", smallest], {n, 1, 4}] (* _Jean-François Alcover, Aug 13 2013 *)
Formula
If A230561(n) exists, then a(n) <= A230561(n) for n > 1, with equality at least for n = 2, and inequality at least for n = 3. - Jonathan Sondow, Oct 24 2013 [Comment edited by N. J. A. Sloane, Apr 03 2021]
Comments