OFFSET
1,1
COMMENTS
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.
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
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
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..360
Daniel J. Bernstein, Detecting perfect powers in essentially linear time, Mathematics of Computation 67 (1998), pp. 1253-1283.
John H. Loxton, Some problems involving powers of integers, Acta Arithmetica 46:2 (1986), pp. 113-123. See Bernstein, Corollary 19.5, for a correction to the proof of Theorem 1.
StackExchange MathOverflow, Are there ever three perfect powers between consecutive squares? Answers by Gjergji Zaimi and Felipe Voloch (2011).
EXAMPLE
27 and 32 are close because they are between 25 and 36.
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 03 2006
STATUS
approved