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

%I A245713 #56 Feb 16 2025 08:33:23
%S A245713 1,8,16,27,32,64,64,81,125,128,216,243,256,256,343,512,512,625,729,
%T A245713 729,1000,1024,1024,1296,1331,1728,2048,2187,2197,2401,2744,3125,3375,
%U A245713 4096,4096,4096,4096,4913,5832,6561,6561,6859,7776,8000,8192,9261
%N A245713 Sorted imperfect powers b^p with b > 0, p > 2, with multiplicity.
%C A245713 No multiple terms for b=1.
%C A245713 This sequence strictly follows requirements of the Beal conjecture.
%C A245713 Less than 550 of these powers satisfy 196 Beal's conjecture equations.
%H A245713 Anatoly E. Voevudko, <a href="/A245713/b245713.txt">Table of n, a(n) for n = 1..11539</a>
%H A245713 American Mathematical Society, <a href="http://www.ams.org/profession/prizes-awards/ams-supported/beal-prize">Beal Prize</a>
%H A245713 Alf van der Poorten, <a href="/A023057/a023057.txt">Remarks on the sequence of 'perfect' numbers</a>
%H A245713 Anatoly E. Voevudko, <a href="/A245713/a245713.txt">Description of all powers in b245713</a>
%H A245713 Eric W. Weisstein, World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Power</a>
%H A245713 Wikipedia, <a href="http://en.wikipedia.org/wiki/Beal%27s_conjecture">Beal's conjecture</a>
%p A245713 N:= 10^5: # to get all terms <= N
%p A245713 L:= [1, seq(seq(b^p, p=3..floor(log[b](N))),b=2..floor(N^(1/3)))]:
%p A245713 sort(L); # _Robert Israel_, Nov 09 2015
%t A245713 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 *)
%o A245713 (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
%Y A245713 Cf. A001597, A023057, A072103, A076467.
%K A245713 nonn,easy
%O A245713 1,2
%A A245713 _Anatoly E. Voevudko_, Jul 30 2014