

A016078


Smallest number that is sum of 2 positive nth powers in 2 different ways.


3




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. 182196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.
Eric Weisstein's World of Mathematics, Diophantine Equation5th Powers


FORMULA

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


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.


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}] (* JeanFrançois 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



