A265731 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, without multiplicity.

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

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 A331701 A145820

Adjacent sequences: A265728 A265729 A265730 * A265732 A265733 A265734

Keyword

nonn,easy

Author

Anatoly E. Voevudko, Dec 14 2015

Status

approved