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 A265732 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, with multiplicity. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS We do not distinguish between the equations C^z = A^x + B^y and C^z = B^y + A^x. This type of equation is used in the Fermat-Catalan conjecture, the ABC conjecture, etc., of course with additional restrictions and conditions. 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 EXAMPLE 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). PROG (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 CROSSREFS Cf. A000290, A245713, A261782, A264901, A265731. Sequence in context: A265221 A286476 A079669 * A047393 A241263 A307417 Adjacent sequences:  A265729 A265730 A265731 * A265733 A265734 A265735 KEYWORD nonn,easy AUTHOR Anatoly E. Voevudko, Dec 14 2015 STATUS approved

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Last modified December 11 21:00 EST 2019. Contains 329937 sequences. (Running on oeis4.)