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%I #18 Jan 22 2021 10:03:07
%S 27,32,125,128,2187,2197,6434856,6436343,312079600999,312079650687,
%T 328080401001,328080696273,11305786504384,11305787424768,
%U 62854898176000,62854912109375,79723529268319,79723537443243,4550858390629024
%N Perfect powers (A001597) that are close, that is, between consecutive squares.
%C It appears that all pairs of close powers involve a cube. For three pairs, the other power is a 7th power. For all remaining pairs, the other power is a 5th power. If this is true, then three powers are never close.
%C For the first 360 terms, 176 pairs are a cube and a 5th power. The remaining four pairs are a cube and a 7th power. - _Donovan Johnson_, Feb 26 2011
%C Loxton proves that the interval [n, n+sqrt(n)] contains at most exp(40 log log n log log log n) powers for n >= 16, and hence there are at most 2*exp(40 log log n log log log n) between consecutive squares in the interval containing n. - _Charles R Greathouse IV_, Jun 25 2017
%H Donovan Johnson, <a href="/A117934/b117934.txt">Table of n, a(n) for n = 1..360</a>
%H Daniel J. Bernstein, <a href="https://cr.yp.to/papers/powers.pdf">Detecting perfect powers in essentially linear time</a>, Mathematics of Computation 67 (1998), pp. 1253-1283.
%H John H. Loxton, <a href="https://eudml.org/doc/205990">Some problems involving powers of integers</a>, Acta Arithmetica 46:2 (1986), pp. 113-123. See Bernstein, Corollary 19.5, for a correction to the proof of Theorem 1.
%H StackExchange MathOverflow, <a href="https://mathoverflow.net/questions/62444/are-there-ever-three-perfect-powers-between-consecutive-squares/62479">Are there ever three perfect powers between consecutive squares?</a> Answers by Gjergji Zaimi and Felipe Voloch (2011).
%e 27 and 32 are close because they are between 25 and 36.
%t nMax=10^14; lst={}; log2Max=Ceiling[Log[2,nMax]]; bases=Table[2,{log2Max}]; powers=bases^Range[log2Max]; powers[[1]]=Infinity; currPP=1; cnt=0; While[nextPP=Min[powers]; nextPP <= nMax, pos=Flatten[Position[powers,nextPP]]; If[MemberQ[pos,2], cnt=0, cnt++ ]; If[cnt>1, AppendTo[lst,{currPP,nextPP}]]; Do[k=pos[[i]]; bases[[k]]++; powers[[k]]=bases[[k]]^k, {i,Length[pos]}]; currPP=nextPP]; Flatten[lst]
%Y Cf. A097056, A117896 (number of perfect powers between consecutive squares n^2 and (n+1)^2).
%Y Cf. A340696, A340700.
%K nonn
%O 1,1
%A _T. D. Noe_, Apr 03 2006
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