A264901 Sorted powers C^z = A^x + B^y with all positive integers and x,y,z > 2, with multiplicity.

16, 32, 64, 64, 128, 128, 128, 243, 256, 256, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024, 1024, 1024, 1024, 2048, 2048, 2048, 2744, 4096, 4096, 4096, 4096, 6561, 6561, 6561, 6561, 8192, 8192, 8192, 8192, 8192, 8192

Offset

1,1

Comments

We do not distinguish between the representations C^z = A^x + B^y and C^z = B^y + A^x.

This sequence is based on the type of equation involved in Beal's conjecture.

Links

Anatoly E. Voevudko, Table of n, a(n) for n = 1..615

American Mathematical Society, Beal Prize

Anatoly E. Voevudko, Description of all powers in b245713

Anatoly E. Voevudko, Description of all powers in b261782

Anatoly E. Voevudko, Description of all powers in b264901

Wikipedia, Beal's conjecture

Example

128 = 64 + 64 ==> 2^7 = 2^6 + 2^6 = 2^6 + 4^3 = 4^3 + 4^3 (but not 4^3 + 2^6).

Prog

(PARI) b264901(lim)=
{my(Lc=List(1), Lb=List(), La=Lb, czn, lan, lbn, lcn, lim2=logint(lim, 2), lim3);
for(z=3, 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)); )); lbn=#Lb; listsort(Lb);
for(i=1, lbn, print(i, " ", Lb[i]))}
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); }

Crossrefs

Cf. A245713, A261782.

Sequence in context: A204645 A236323 A018923 * A339840 A172418 A369170

Adjacent sequences: A264898 A264899 A264900 * A264902 A264903 A264904

Keyword

nonn,easy

Author

Anatoly E. Voevudko, Nov 28 2015

Status

approved