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A088677 Numbers that can be represented as j^6 + k^6, with 0 < j < k, in exactly one way.

%I A088677 #21 Feb 16 2025 08:32:51
%S A088677 65,730,793,4097,4160,4825,15626,15689,16354,19721,46657,46720,47385,
%T A088677 50752,62281,117650,117713,118378,121745,133274,164305,262145,262208,
%U A088677 262873,266240,277769,308800,379793,531442,531505,532170,535537,547066
%N A088677 Numbers that can be represented as j^6 + k^6, with 0 < j < k, in exactly one way.
%C A088677 Conjecture: no number can be expressed as such a sum in more than one way.
%C A088677 Ekl (1996) has searched and found no solutions to the 6.2.2 Diophantine equation A^6 + B^6 = C^6 + D^6 with sums less than 7.25 * 10^26. - _Jonathan Vos Post_, May 04 2006
%H A088677 R. L. Ekl, <a href="http://dx.doi.org/10.1090/S0025-5718-98-00979-X">New Results in Equal Sums of Like Powers</a>, Math. Comput. 67, 1309-1315, 1998.
%H A088677 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation6thPowers.html">Diophantine Equation: 6th Powers</a>.
%e A088677 65 = 1^6 + 2^6.
%t A088677 lst={};e=6;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,2*8!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 22 2009 *)
%o A088677 (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) }
%Y A088677 Cf. A003358, A088719 (7th powers).
%K A088677 nonn
%O A088677 1,1
%A A088677 _Cino Hilliard_, Nov 22 2003
%E A088677 Edited by _Don Reble_, May 03 2006