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

1,1

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: A301659 A265221 A286476 * A047393 A241263 A307417

Adjacent sequences: A265729 A265730 A265731 * A265733 A265734 A265735

Keyword

nonn,easy

Author

Anatoly E. Voevudko, Dec 14 2015

Status

approved