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%I #34 Jun 28 2023 14:42:07
%S 33,244,275,1025,1056,1267,3126,3157,3368,4149,7777,7808,8019,8800,
%T 10901,16808,16839,17050,17831,19932,24583,32769,32800,33011,33792,
%U 35893,40544,49575,59050,59081,59292,60073,62174,66825,75856,91817
%N Numbers of form x^5 + y^5, x,y > 0 and x <> y.
%C 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.
%D Guy, Richard K., Unsolved Problems in Number Theory, 2nd Ed., Springer-Verlag(1994), pp. 140.
%H David A. Corneth, <a href="/A088703/b088703.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Generalized_taxicab_number">Generalized Taxicab Numbers</a>
%e 33 = 2^5 + 1^5, so 33 is in sequence. 64 = 2^5 + 2^5 is not.
%t 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 *)
%t Union[#[[1]]^5+#[[2]]^5&/@Subsets[Range[10],{2}]] (* _Harvey P. Dale_, Apr 25 2012 *)
%o (PARI) powers2(m1,m2,p1) = { for(k=m1,m2, a=powers(k,p1); if(a==1,print1(k", ")) ); }
%o 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) }
%Y Subsequence of A003347.
%Y Cf. A088687 (4th powers), A088677 (6th powers), A046881 (bounds for double reps).
%K nonn
%O 1,1
%A _Cino Hilliard_, Nov 22 2003
%E Edited by _Ralf Stephan_, Dec 30 2004
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