A265732 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, with multiplicity.
8, 9, 16, 16, 25, 25, 32, 32, 32, 36, 36, 64, 64, 64, 81, 81, 100, 100, 100, 125, 125, 128, 128, 128, 128, 128, 128, 144, 144, 169, 196, 225, 225, 225, 225, 243, 256, 256, 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
Offset: 1
Examples
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).
Links
- Anatoly E. Voevudko, Table of n, a(n) for n = 1..16865
- Anatoly E. Voevudko, Description of all powers in b265732
- Anatoly E. Voevudko, Description of all powers in b265731
- Anatoly E. Voevudko, Description of all powers in b245713
- Anatoly E. Voevudko, Description of all powers in b261782
- Wikipedia, abc conjecture
- Wikipedia, Fermat-Catalan conjecture
Programs
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PARI
A265732(lim,bflag=0)= {my(Lc=List(1),Lb=List(),La=Lb,czn,lcn,lan,lim2=logint(lim,2),lim3,k); for(z=2,lim2,lim3=sqrtnint(lim,z); for(C=2,lim3,listput(Lc,C^z)) ); lcn = #Lc; if(lcn==0,return(-1)); for(i=1,lcn, for(j=i,lcn, czn=Lc[i]+Lc[j]; if(czn>lim, next); La=findinlista(Lc, czn); lan=#La; if(!lan, next); for(k=1,lan, listput(Lb, czn)))); lcn=#Lb; listsort(Lb,0); if(bflag,for(i=1,lcn,print(i ," ",Lb[i]))); if(!bflag,return(Vec(Lb))); } findinlista(list, item, sind=1)={my(ln=#list, Li=List()); if(ln==0||sind<1||sind>ln, return(Li)); for(i=sind, ln, if(list[i]==item, listput(Li,i))); return(Li); } \\ Anatoly E. Voevudko, Nov 23 2015
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