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%I #15 Jan 09 2016 00:32:02
%S 8,9,16,16,25,25,32,32,32,36,36,64,64,64,81,81,100,100,100,125,125,
%T 128,128,128,128,128,128,144,144,169,196,225,225,225,225,243,256,256,
%U 256,289,289,289,324,324,324,343,400,400,400,441,512,512,512,512,512,512,512,512,512,512,512,512,512,512,576
%N Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, with multiplicity.
%C We do not distinguish between the equations C^z = A^x + B^y and C^z = B^y + A^x.
%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="/A265732/b265732.txt">Table of n, a(n) for n = 1..16865</a>
%H Anatoly E. Voevudko, <a href="/A265732/a265732.txt">Description of all powers in b265732</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 128 = 64 + 64 ==> 2^7 = 8^2 + 8^2 = 8^2 + 4^3 = 8^2 + 2^6 = 4^3 + 4^3 = 4^3 + 2^6 = 2^6 + 2^6 (but not 4^3 + 8^2, 2^6 + 8^2, 2^6 + 4^3).
%o (PARI) A265732(lim,bflag=0)=
%o {my(Lc=List(1),Lb=List(),La=Lb,czn,lcn,lan,lim2=logint(lim,2),lim3,k);
%o for(z=2,lim2,lim3=sqrtnint(lim,z); for(C=2,lim3,listput(Lc,C^z)) );
%o lcn = #Lc; if(lcn==0,return(-1));
%o for(i=1,lcn, for(j=i,lcn, czn=Lc[i]+Lc[j]; if(czn>lim, next);
%o La=findinlista(Lc, czn); lan=#La; if(!lan, next);
%o for(k=1,lan, listput(Lb, czn)))); lcn=#Lb; listsort(Lb,0);
%o if(bflag,for(i=1,lcn,print(i ," ",Lb[i]))); if(!bflag,return(Vec(Lb)));
%o }
%o findinlista(list, item, sind=1)={my(ln=#list, Li=List());
%o if(ln==0||sind<1||sind>ln, return(Li));
%o for(i=sind, ln, if(list[i]==item, listput(Li,i))); return(Li);
%o } \\ _Anatoly E. Voevudko_, Nov 23 2015
%Y Cf. A000290, A245713, A261782, A264901, A265731.
%K nonn,easy
%O 1,1
%A _Anatoly E. Voevudko_, Dec 14 2015
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