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

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

Author

Anatoly E. Voevudko, Jul 30 2014

Keywords

• nonn
• easy

Comments

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.

Links

• 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

Crossrefs

Cf. A001597, A023057, A072103, A076467.

Programs

Maple

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

Mathematica

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 *)

PARI

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