A016078 Smallest number that is sum of 2 positive n-th powers in 2 different ways.

4, 50, 1729, 635318657

Offset

1,1

Comments

If it exists, a(5) > 1.02*10^26 (see eqn. (27) at the Mathworld link). - Jon E. Schoenfield, Jan 05 2019

Links

Table of n, a(n) for n=1..4.

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

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]

Example

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).

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 *)

Crossrefs

Cf. A046881, A230561.

Sequence in context: A201829 A201209 A026865 * A327229 A231832 A193157

Adjacent sequences: A016075 A016076 A016077 * A016079 A016080 A016081

Keyword

nonn,nice,hard,more

Author

Robert G. Wilson v, Dec 11 1999

Status

approved