A097056 -id:A097056 - cp's OEIS frontend

A097054 Nonsquare perfect powers.

8, 27, 32, 125, 128, 216, 243, 343, 512, 1000, 1331, 1728, 2048, 2187, 2197, 2744, 3125, 3375, 4913, 5832, 6859, 7776, 8000, 8192, 9261, 10648, 12167, 13824, 16807, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653

Offset: 1

Hugo Pfoertner, Jul 21 2004

Terms of A001597 that are not in A000290.

All terms of this sequence are also in A070265 (odd powers), but omitting those odd powers that are also a square (e.g. 64=4^3=8^2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Power.
Eric Weisstein's World of Mathematics, Odd Power.

Cf. A001597 (perfect powers), A000290 (the squares), A008683, A070265 (odd powers), A097055, A097056, A239870, A239728, A093771.

import Data.Map (singleton, findMin, deleteMin, insert)
a097054 n = a097054_list !! (n-1)
a097054_list = f 9 (3, 2) (singleton 4 (2, 2)) where
   f zz (bz, be) m
    | xx < zz && even be =
                f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
    | xx < zz = xx :
                f zz (bz, be+1) (insert (bx*xx) (bx, be+1) $ deleteMin m)
    | xx > zz = f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)
    | otherwise = f (zz + 2 * bz + 1) (bz + 1, 2) m
    where (xx, (bx, be)) = findMin m
-- Reinhard Zumkeller, Mar 28 2014

# uses code of A001597
for n from 4 do
    if not issqr(n) and isA001597(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jan 13 2021

nn = 50653; Select[Union[Flatten[Table[n^i, {i, Prime[Range[2, PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]], ! IntegerQ[Sqrt[#]] &] (* T. D. Noe, Apr 19 2011 *)

is(n)=ispower(n)%2 \\ Charles R Greathouse IV, Aug 28 2016

list(lim)=my(v=List()); forprime(e=3,logint(lim\=1,2), for(b=2,sqrtnint(lim,e), if(!issquare(b), listput(v,b^e)))); Set(v) \\ Charles R Greathouse IV, Jan 09 2023

from sympy import mobius, integer_nthroot
def A097054(n):
    def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(3,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

A052409(a(n)) is odd. - Reinhard Zumkeller, Mar 28 2014

Sum_{n>=1} 1/a(n) = 1 - zeta(2) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 0.2295303015... - Amiram Eldar, Dec 21 2020

A340700 Lower of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

27, 64, 125, 243, 1000, 1296, 2187, 50625, 59049, 194481, 279841, 456533, 614125, 3111696, 6434856, 22665187, 25411681, 38950081, 62742241, 96059601, 131079601, 418161601, 506250000, 741200625, 796594176, 1249198336, 2136719872, 2217342464, 5554571841, 5802782976

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

It is conjectured that the intersection of A340700 and A340701 is empty, i.e., that no 3 immediately consecutive perfect powers with all exponents > 2 (A076467) exist. No counterexample < 3.4*10^30 was found.

			Initial terms of sequences A340700 .. A340706:
a(n) = x^p,
A340701(n) = A340703(n)^A340705(n) = y^q,
A340706(n) = A340701(n) - a(n) = y^q - x^p.
.
  n  a(n)    x ^  p  A340701    y ^  q  A340706 adjacent squares
  1    27 =  3 ^  3,      32 =  2 ^  5,      5  5^2=25, 6^2=36
  2    64 =  2 ^  6,      81 =  3 ^  4,     17  8^2=64, 9^2=81
  3   125 =  5 ^  3,     128 =  2 ^  7,      3  11^2=121, 12^2=144
  4   243 =  3 ^  5,     256 =  2 ^  8,     13  15^2=225, 16^2=256
  5  1000 = 10 ^  3,    1024 =  2 ^ 10,     24  31^2=961, 32^2=1024
  6  1296 =  6 ^  4,    1331 = 11 ^  3,     35  36^2=1296, 37^2=1369
  7  2187 =  3 ^  7,    2197 = 13 ^  3,     10  46^2=2116, 47^2=2209
  8 50625 = 15 ^  4,   50653 = 37 ^  3,     28  225^2=50625, 226^2=51076
  9 59049 =  3 ^ 10,   59319 = 39 ^  3,    270  243^2=59049, 244^2=59536

Hugo Pfoertner, Table of n, a(n) for n = 1..1670
StackExchange MathOverflow, Are there ever three perfect powers between consecutive squares? Answers by Gjergji Zaimi and Felipe Voloch (2011).
Michel Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 27 Aug 2009.

The corresponding upper members of the pairs are A340701.
Cf. A001597, A076467, A097056, A340642, A340702, A340703, A340704, A340705, A340706.
Cf. A117934 (excluding pairs where one of the members is a square).

a(n) = A340702(n)^A340704(n) = A340701(n) - A340706(n).

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

Original entry on oeis.org

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]

A097055 Numbers n such that the interval n^2 < x < (n+1)^2 contains at least one nonsquare perfect power A097054.

Original entry on oeis.org

2, 5, 11, 14, 15, 18, 22, 31, 36, 41, 45, 46, 52, 55, 58, 70, 76, 82, 88, 89, 90, 96, 103, 110, 117, 129, 132, 140, 148, 156, 164, 172, 181, 189, 198, 207, 225, 234, 243, 252, 262, 272, 279, 281, 291, 301, 311, 316, 322, 332, 353, 362, 364, 374, 385, 396, 401

Offset: 1

Hugo Pfoertner, Jul 21 2004

nonn

			a(1)=2 because 2^2<2^3<3^2, a(2)=5: 5^2<3^3<2^5<6^2, a(3)=11: 11^2<5^3<2^7<12^2.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000290, A097054, A097056.

nn=1000^2; pp=Select[Union[Flatten[Table[n^i, {i,Prime[Range[2,PrimePi[Log[2,nn]]]]}, {n,2,nn^(1/i)}]]], !IntegerQ[Sqrt[#]]&]; Union[Floor[Sqrt[pp]]] (* T. D. Noe, Apr 19 2011 *)

is(n)=my(n2=n^2,t=n2+2*n); for(e=3,logint(t,2), if(sqrtnint(t,e)^e>n2, return(1))); 0 \\ Charles R Greathouse IV, Aug 28 2016

from itertools import count, islice
from sympy import mobius, integer_nthroot
def A097055_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n: sum(mobius(k)*(1-integer_nthroot((n+1)**2,k)[0]) for k in range((n**2).bit_length(),((n+1)**2).bit_length()))+sum(mobius(k)*(integer_nthroot(n**2,k)[0]-integer_nthroot((n+1)**2,k)[0]) for k in range(3,(n**2).bit_length())), count(max(startvalue,2)))
A097055_list = list(islice(A097055_gen(),30)) # Chai Wah Wu, Aug 14 2024

a(n) ~ n^(3/2). - Charles R Greathouse IV, Aug 28 2016

A117896 Number of perfect powers between consecutive squares n^2 and (n+1)^2.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

T. D. Noe, Mar 31 2006, Feb 15 2010

nonn

a(n)=2 only 14 times for n^2 < 2^63. What is the least n such that a(n)=3? Is a(n) bounded?

			a(5)=2 because powers 27 and 32 are between 25 and 36.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. H. Loxton, Some problems involving powers of integers, Acta Arith., 46 (1986), pp. 113-123.
C. L. Stewart, On heights of multiplicatively dependent algebraic numbers, Acta Arith. 133 (2008), pp. 97-108.
J. Turk, Multiplicative properties of integers in short intervals, Proc. Kon. Ned. Akad. Wet. (A) 83 (1980), pp. 429-436.

Cf. A001597 (perfect powers), A014085 (primes between squares), A097055, A097056, A117934.

nn=151^2; powers=Join[{1}, Union[Flatten[Table[n^i, {i,Prime[Range[PrimePi[Log[2,nn]]]]}, {n,2,nn^(1/i)}]]]]; t=Table[0,{Sqrt[nn]-1}]; Do[n=Floor[Sqrt[i]]; If[i>n^2, t[[n]]++], {i,powers}]; t (* revised, T. D. Noe, Apr 19 2011 *)

a(n)=my(k);-sum(e=3,2*log(n+1)\log(2),k=round((n+1/2)^(2/e))^e;if(n^2Charles R Greathouse IV, Dec 19 2011

Trivially, a(n) << log n/log log n. Turk gives a(n) << sqrt(log n) and Loxton improves this to a(n) <= exp(40 sqrt(log log n log log log n)). Stewart improves the constant from 40 to 30 and conjectures that a(n) < 3 for all but finitely many n. - Charles R Greathouse IV, Dec 11 2012

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

A289690 Least k such that there are exactly n perfect powers between 10k and 10k + 10.

Original entry on oeis.org

5, 1, 2, 12, 0

Offset: 0

Wolfram Hüttermann, Jul 09 2017

I do not know if a(5) exists. If it does, the numbers 10k+1, 10k+3, 10k+5, 10k+7, 10k+9 will be perfect powers. But those numbers are very scarce.
Further, a(6), ..., a(10) cannot exist because of the Mihailescu theorem, as the only adjoining perfect powers are 8 and 9.
a(5) is extremely unlikely to exist; if it does, it is larger than 10^70. - Charles R Greathouse IV, Jul 21 2017

			If n=2, then there are 2 power numbers between 20 and 30: 25 and 27, and this is the least k with this property.

Cf. A001597, A097056.

a(n)=my(k=0); while(sum(j=10*k+1, 10*k+9, (j==1) || ispower(j)) !=n, k++); k; \\ Michel Marcus, Jul 20 2017

cp's OEIS Frontend

A097054 Nonsquare perfect powers.

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A340700 Lower of a pair of adjacent perfect powers, both with exponents > 2.

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

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A097055 Numbers n such that the interval n^2 < x < (n+1)^2 contains at least one nonsquare perfect power A097054.

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A117896 Number of perfect powers between consecutive squares n^2 and (n+1)^2.

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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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A289690 Least k such that there are exactly n perfect powers between 10k and 10k + 10.

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