A117896 -id:A117896 - cp's OEIS frontend

A117934 Perfect powers (A001597) that are close, that is, between consecutive squares.

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

Offset: 1

T. D. Noe, Apr 03 2006

nonn

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

			27 and 32 are close because they are between 25 and 36.

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).

Cf. A097056, A117896 (number of perfect powers between consecutive squares n^2 and (n+1)^2).

Cf. A340696, A340700.

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]

A173341 Numbers n such that n^5 and a cube are between consecutive squares.

Original entry on oeis.org

2, 23, 199, 201, 408, 575, 603, 1354, 1628, 4995, 5745, 7320, 7994, 12634, 42637, 44217, 45962, 67132, 82131, 82351, 91116, 91134, 146521, 177682, 229863, 359373, 394826, 458908, 462763, 512012, 665719, 728982, 1009965, 1156978, 1450803

Offset: 1

T. D. Noe, Feb 16 2010

nonn

This sequence appears to be infinite. Sequence A117594 is a subsequence. The corresponding sequence for n^7 is A173342. Are there ever more than two perfect powers between consecutive squares?

			2 is here because 2^5=32 and 3^3=27 are between 5^2=25 and 6^2=36.
23 is here because 23^5 and 186^3 are between 2536^2 and 2537^2.
199 is here because 199^5 and 6783^3 are between 558640^2 and 558641^2.

A097056, A117896

t={}; Do[n2=Floor[n^(5/2)]; n3=Round[n^(5/3)]; If[n2^2

A173342 Numbers n such that n^7 and a cube are between consecutive squares.

Original entry on oeis.org

2, 3, 498, 2266144, 272585923

Offset: 1

T. D. Noe, Feb 16 2010

nonn

No other terms < 10^8. The corresponding sequence for n^5 is A173341. Are there ever more than two perfect powers between consecutive squares?

a(6) > 10^10. [From Donovan Johnson, Apr 17 2010]

			2 is here because 2^7=128 and 5^3=125 are between 11^2=121 and 12^2=144.
3 is here because 3^7=2187 and 13^3=2197 are between 46^2=2116 and 47^2=2209.
498 is here because 498^7 and 1965781^3 are between 2756149047^2 and 2756149048^2.
2266144 is here because 2266144^7 and 674534510965903^3 are between 17518876914709436673663^2 and 17518876914709436673664^2.

A097056, A117896

t={}; Do[n2=Floor[n^(7/2)]; n3=Round[n^(7/3)]; If[n2^2

a(5) from Donovan Johnson, Apr 17 2010

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A117934 Perfect powers (A001597) that are close, that is, between consecutive squares.

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A173341 Numbers n such that n^5 and a cube are between consecutive squares.

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A173342 Numbers n such that n^7 and a cube are between consecutive squares.

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