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%I #15 Jan 09 2016 00:26:42
%S 8,9,16,25,32,36,64,81,100,125,128,144,169,196,225,243,256,289,324,
%T 343,400,441,512,576,625,676,784,841,900,1000,1024,1089,1156,1225,
%U 1296,1369,1521,1600,1681,1728,1764,1849,2025,2048,2197,2304,2500,2601,2704,2744,2809,2916,3025,3125
%N Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, without multiplicity.
%C This type of equation is used in the Fermat-Catalan conjecture, the ABC conjecture, etc., of course, with additional restrictions and conditions.
%H Anatoly E. Voevudko, <a href="/A265731/b265731.txt">Table of n, a(n) for n = 1..7253</a>
%H Anatoly E. Voevudko, <a href="/A265731/a265731.txt">Description of all powers in b265731</a>
%H Anatoly E. Voevudko, <a href="/A245713/a245713.txt">Description of all powers in b245713</a>
%H Anatoly E. Voevudko, <a href="/A261782/a261782.txt">Description of all powers in b261782</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fermat%E2%80%93Catalan_conjecture">Fermat-Catalan conjecture</a>
%e 8 = 2^3 = 2^2 + 2^2; 9 = 3^2 = 1^3 + 2^3; 16 = 4^2 = 2^3 + 2^3; etc.
%o (PARI) A265731(lim,bflag=0)={my(Lcz=List(1),Lb=List(),czn,lczn,lbn,lim2=logint(lim, 2),lim3);
%o for(z=2, lim2, lim3=sqrtnint(lim, z); for(C=2, lim3, listput(Lcz, C^z)));
%o Lcz=Set(Lcz); lczn = #Lcz; if(lczn==0,return(-1));
%o for(i=1, lczn, for(j=i, lczn, czn=Lcz[i]+Lcz[j]; if(czn>lim, break);
%o if(setsearch(Lcz, czn), listput(Lb, czn)))); listsort(Lb,1); lbn=#Lb;
%o if(bflag, for(i=1,lbn,print(i , " ", Lb[i]))); if(!bflag,return(Vec(Lb))); }
%o \\ _Anatoly E. Voevudko_, Nov 23 2015
%Y Cf. A000290, A245713, A261782, A264901, A265732.
%K nonn,easy
%O 1,1
%A _Anatoly E. Voevudko_, Dec 14 2015
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