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 A265731 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, without multiplicity. 2
 8, 9, 16, 25, 32, 36, 64, 81, 100, 125, 128, 144, 169, 196, 225, 243, 256, 289, 324, 343, 400, 441, 512, 576, 625, 676, 784, 841, 900, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1521, 1600, 1681, 1728, 1764, 1849, 2025, 2048, 2197, 2304, 2500, 2601, 2704, 2744, 2809, 2916, 3025, 3125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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..7253 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 8 = 2^3 = 2^2 + 2^2; 9 = 3^2 = 1^3 + 2^3; 16 = 4^2 = 2^3 + 2^3; etc. PROG (PARI) A265731(lim, bflag=0)={my(Lcz=List(1), Lb=List(), czn, lczn, lbn, lim2=logint(lim, 2), lim3); for(z=2, lim2, lim3=sqrtnint(lim, z); for(C=2, lim3, listput(Lcz, C^z))); Lcz=Set(Lcz); lczn = #Lcz; if(lczn==0, return(-1)); for(i=1, lczn, for(j=i, lczn, czn=Lcz[i]+Lcz[j]; if(czn>lim, break); if(setsearch(Lcz, czn), listput(Lb, czn)))); listsort(Lb, 1);  lbn=#Lb; if(bflag, for(i=1, lbn, print(i , " ", Lb[i]))); if(!bflag, return(Vec(Lb))); } \\ Anatoly E. Voevudko, Nov 23 2015 CROSSREFS Cf. A000290, A245713, A261782, A264901, A265732. Sequence in context: A227649 A227648 A192636 * A227646 A145820 A227647 Adjacent sequences:  A265728 A265729 A265730 * A265732 A265733 A265734 KEYWORD nonn,easy AUTHOR Anatoly E. Voevudko, Dec 14 2015 STATUS approved

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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)