A088703 Numbers of form x^5 + y^5, x,y > 0 and x <> y.

33, 244, 275, 1025, 1056, 1267, 3126, 3157, 3368, 4149, 7777, 7808, 8019, 8800, 10901, 16808, 16839, 17050, 17831, 19932, 24583, 32769, 32800, 33011, 33792, 35893, 40544, 49575, 59050, 59081, 59292, 60073, 62174, 66825, 75856, 91817

Offset: 1

Author

Cino Hilliard, Nov 22 2003

Keywords

• nonn

Comments

Up to n = 100000, no instances occur where n is the sum of two distinct 5th powers in two different ways. Conjecture: no number can be expressed as the sum of two 5th powers in more than one way: A046881.

Examples

			33 = 2^5 + 1^5, so 33 is in sequence. 64 = 2^5 + 2^5 is not.
		

References

• Guy, Richard K., Unsolved Problems in Number Theory, 2nd Ed., Springer-Verlag(1994), pp. 140.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Wikipedia, Generalized Taxicab Numbers

Crossrefs

Subsequence of A003347.

Cf. A088687 (4th powers), A088677 (6th powers), A046881 (bounds for double reps).

Programs

Mathematica

lst={};e=5;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,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Union[#[[1]]^5+#[[2]]^5&/@Subsets[Range[10],{2}]] (* Harvey P. Dale, Apr 25 2012 *)

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) }

Extensions

Edited by Ralf Stephan, Dec 30 2004