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A264901 Sorted powers C^z = A^x + B^y with all positive integers and x,y,z > 2, with multiplicity.

%I A264901 #22 Jan 09 2016 00:41:30
%S A264901 16,32,64,64,128,128,128,243,256,256,512,512,512,512,512,512,1024,
%T A264901 1024,1024,1024,1024,1024,2048,2048,2048,2744,4096,4096,4096,4096,
%U A264901 6561,6561,6561,6561,8192,8192,8192,8192,8192,8192
%N A264901 Sorted powers C^z = A^x + B^y with all positive integers and x,y,z > 2, with multiplicity.
%C A264901 We do not distinguish between the representations C^z = A^x + B^y and C^z = B^y + A^x.
%C A264901 This sequence is based on the type of equation involved in Beal's conjecture.
%H A264901 Anatoly E. Voevudko, <a href="/A264901/b264901.txt">Table of n, a(n) for n = 1..615</a>
%H A264901 American Mathematical Society, <a href="http://www.ams.org/profession/prizes-awards/ams-supported/beal-prize">Beal Prize</a>
%H A264901 Anatoly E. Voevudko, <a href="/A245713/a245713.txt">Description of all powers in b245713</a>
%H A264901 Anatoly E. Voevudko, <a href="/A261782/a261782.txt">Description of all powers in b261782</a>
%H A264901 Anatoly E. Voevudko, <a href="/A264901/a264901_1.txt">Description of all powers in b264901</a>
%H A264901 Wikipedia, <a href="http://en.wikipedia.org/wiki/Beal%27s_conjecture">Beal's conjecture</a>
%e A264901 128 = 64 + 64 ==> 2^7 = 2^6 + 2^6 = 2^6 + 4^3 = 4^3 + 4^3 (but not 4^3 + 2^6).
%o A264901 (PARI) b264901(lim)=
%o A264901 {my(Lc=List(1),Lb=List(),La=Lb,czn,lan,lbn,lcn,lim2=logint(lim,2),lim3);
%o A264901 for(z=3,lim2,lim3=sqrtnint(lim, z); for(C=2,lim3, listput(Lc, C^z)));
%o A264901 lcn=#Lc; if(lcn==0, return(-1));
%o A264901 for(i=1,lcn, for(j=i,lcn, czn=Lc[i]+Lc[j]; if(czn>lim, next);
%o A264901 La=findinlista(Lc,czn); lan=#La; if(!lan, next);
%o A264901 for(k=1,lan, listput(Lb, czn));)); lbn=#Lb; listsort(Lb);
%o A264901 for(i=1,lbn, print(i," ",Lb[i]))}
%o A264901 findinlista(list, item, sind=1)=
%o A264901 {my(ln=#list, Li=List()); if(ln==0 || sind<1 || sind>ln, return(Li));
%o A264901 for(i=sind, ln, if(list[i]==item, listput(Li,i))); return(Li);}
%Y A264901 Cf. A245713, A261782.
%K A264901 nonn,easy
%O A264901 1,1
%A A264901 _Anatoly E. Voevudko_, Nov 28 2015