A088719 Numbers that can be represented as a^7 + b^7, with 0 < a < b, in exactly one way.

129, 2188, 2315, 16385, 16512, 18571, 78126, 78253, 80312, 94509, 279937, 280064, 282123, 296320, 358061, 823544, 823671, 825730, 839927, 901668, 1103479, 2097153, 2097280, 2099339, 2113536, 2175277, 2377088, 2920695

Offset

1,1

Comments

Conjecture: no number can be expressed as such a sum in more than one way.

No solutions to the 7.2.2 (A^7 + B^7 = C^7 + D^7), 7.2.3, 7.2.4, or 7.2.5 equations are known. The smallest 7.2.6 equation is: 125^7 + 24^7 = 121^7 + 94^7 + 83^7 + 61^7 + 57^7 + 27^7 = 476841744674549. - Jonathan Vos Post, May 04 2006

References

Sastry, S. and Rai, T. "On Equal Sums of Like Powers." Math. Student 16, 18-19, 1948.

Links

Table of n, a(n) for n=1..28.

R. L. Ekl, Equal Sums of Four Seventh Powers, Math. Comput. 65, 1755-1756, 1996.

R. L. Ekl, New Results in Equal Sums of Like Powers, Math. Comput. 67, 1309-1315, 1998.

Eric Weisstein's World of Mathematics, Diophantine Equation: 7th Powers

Example

129 = 1^7+2^7.

Mathematica

lst={}; e=7; Do[Do[x=a^e; Do[y=b^e; If[x+y==n, AppendTo[lst, n]], {b, Floor[(n-x)^(1/e)], a+1, -1}], {a, Floor[n^(1/e)], 1, -1}], {n, 3*8!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 23 2009 *)

Prog

(PARI) powers2(m1, m2, p1) = { for(k=m1, m2, a=powers(k, p1); if(a==1, print1(k", ")) ); } powers(n, p) = { z1=0; z2=0; c=0; cr = floor(n^(1/p)+1); for(x=1, cr, for(y=x+1, cr, z1=x^p+y^p; if(z1 == n, c++); ); ); return(c) }

Crossrefs

Cf. A003369, A155468 (8th powers).

Sequence in context: A240417 A353940 A351270 * A321563 A034681 A351302

Adjacent sequences: A088716 A088717 A088718 * A088720 A088721 A088722

Keyword

nonn

Author

Cino Hilliard, Nov 22 2003

Extensions

Edited by Don Reble, May 03 2006

Status

approved