Anatoly E. Voevudko - cp's OEIS frontend

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

Anatoly E. Voevudko, Nov 28 2015

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.

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

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

Cf. A245713, A261782.

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);}

A265732 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, with multiplicity.

Original entry on oeis.org

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

Anatoly E. Voevudko, Dec 14 2015

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.

			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).

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

Cf. A000290, A245713, A261782, A264901, A265731.

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

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

Original entry on oeis.org

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

Anatoly E. Voevudko, Dec 14 2015

This type of equation is used in the Fermat-Catalan conjecture, the ABC conjecture, etc., of course, with additional restrictions and conditions.

			8 = 2^3 = 2^2 + 2^2; 9 = 3^2 = 1^3 + 2^3; 16 = 4^2 = 2^3 + 2^3; etc.

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

Cf. A000290, A245713, A261782, A264901, A265732.

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

A261782 Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.

Original entry on oeis.org

16, 32, 64, 128, 243, 256, 512, 1024, 2048, 2744, 4096, 6561, 8192, 16384, 32768, 65536, 131072, 177147, 185193, 262144, 474552, 524288, 614656, 810000, 941192, 1048576, 1124864, 1419857, 1500625, 2097152, 3241792, 4194304

Offset: 1

Anatoly E. Voevudko, Aug 31 2015

nonn

Beal's conjecture states that A, B, and C have a common prime factor.

Theorem. If A, B are odd and x, y are even, Beal's conjecture has no counterexample. Proof: Let D be odd, D > 1 and let w be even, w > 2. Then D^w == 9 (mod 24) while D == 0 (mod 3); otherwise, D^w == 1 (mod 24) (trivial). Any even C^z == {0; 8; 16} (mod 24): if C == 0 (mod 3), C^z == 0 (mod 24); if C == 1 (mod 3), C^z == 16 (mod 24); if C == 2 (mod 3), C^z == 8 (mod 24), while z is odd, and C^z == 16 (mod 24), while z is even (trivial). But C^z == (x'+y') (mod 24) where A^x = x' (mod 24), B^y = y' (mod 24); since (x'+y') = {2; 10; 18}, C^z == {2; 10; 18} (mod 24), which cannot be a counterexample to Beal's conjecture. - Sergey Pavlov, May 08 2021

			2^3 + 2^3 = 2^4 = 16, so 16 is in the sequence.

Anatoly E. Voevudko and Charles R Greathouse IV, Table of n, a(n) for n = 1..1229 (first 196 terms from Voevudko)
American Mathematical Society, Beal Prize
Anatoly E. Voevudko, Description of all powers in b245713
Anatoly E. Voevudko, Description of all powers in b261782
Wikipedia, Beal's conjecture

Subsequence of A076467.

Cf. A245713.

is(n)=if(ispower(n)<3, return(0)); for(x=3,logint((n+1)\2,2), for(A=2,sqrtnint(n,x), if(ispower(n-A^x)>2, return(1)))); 0 \\ Charles R Greathouse IV, Sep 03 2015

list(lim)=my(v=List(),u=v,t); for(z=3,logint(lim\=1,2), for(C=2,sqrtnint(lim,z), listput(v,C^z))); v=Set(v); for(i=1,#v, for(j=i,#v, t=v[i]+v[j]; if(t>lim, break); if(setsearch(v,t), listput(u,t)))); Set(u) \\ Charles R Greathouse IV, Sep 03 2015

A245713 Sorted imperfect powers b^p with b > 0, p > 2, with multiplicity.

Original entry on oeis.org

1, 8, 16, 27, 32, 64, 64, 81, 125, 128, 216, 243, 256, 256, 343, 512, 512, 625, 729, 729, 1000, 1024, 1024, 1296, 1331, 1728, 2048, 2187, 2197, 2401, 2744, 3125, 3375, 4096, 4096, 4096, 4096, 4913, 5832, 6561, 6561, 6859, 7776, 8000, 8192, 9261

Offset: 1

Anatoly E. Voevudko, Jul 30 2014

No multiple terms for b=1.
This sequence strictly follows requirements of the Beal conjecture.
Less than 550 of these powers satisfy 196 Beal's conjecture equations.

Anatoly E. Voevudko, Table of n, a(n) for n = 1..11539
American Mathematical Society, Beal Prize
Alf van der Poorten, Remarks on the sequence of 'perfect' numbers
Anatoly E. Voevudko, Description of all powers in b245713
Eric W. Weisstein, World of Mathematics, Perfect Power
Wikipedia, Beal's conjecture

Cf. A001597, A023057, A072103, A076467.

N:= 10^5: # to get all terms <= N
L:= [1, seq(seq(b^p, p=3..floor(log[b](N))),b=2..floor(N^(1/3)))]:
sort(L); # Robert Israel, Nov 09 2015

mx = 10000; Join[{1}, Sort@ Flatten@ Table[b^p, {b, 2, Sqrt@ mx}, {p, 3, Log[b, mx]}]] (* Robert G. Wilson v, Nov 09 2015 *)

A245713(lim)={my(L=List(1),lim2=logint(lim,2));for(p=3,lim2, for(b=2,sqrtnint(lim,p),listput(L, b^p);));listsort(L); print(L);} \\ Anatoly E. Voevudko, Sep 21 2015

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

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A265732 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, with multiplicity.

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

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A261782 Powers C^z = A^x + B^y with positive integers A,B,C,x,y,z such that x,y,z > 2.

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A245713 Sorted imperfect powers b^p with b > 0, p > 2, with multiplicity.

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