

A117934


Perfect powers (A001597) that are close, that is, between consecutive squares.


4



27, 32, 125, 128, 2187, 2197, 6434856, 6436343, 312079600999, 312079650687, 328080401001, 328080696273, 11305786504384, 11305787424768, 62854898176000, 62854912109375, 79723529268319, 79723537443243, 4550858390629024
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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


REFERENCES

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. 12531283.
John H. Loxton, <a href="https://eudml.org/doc/205990">Some problems involving powers of integers</a>, Acta Arithmetica 46:2 (1986), pp. 113123. See Bernstein, Corollary 19.5, for a correction to the proof of Theorem 1.


LINKS

Donovan Johnson, Table of n, a(n) for n = 1..360


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

Cf. A117896 (number of perfect powers between consecutive squares n^2 and (n+1)^2).
Sequence in context: A275188 A198147 A144862 * A173136 A030134 A290842
Adjacent sequences: A117931 A117932 A117933 * A117935 A117936 A117937


KEYWORD

nonn


AUTHOR

T. D. Noe, Apr 03 2006


STATUS

approved



