A097055 -id:A097055 - 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

A097056 Numbers n such that the interval n^2 < x < (n+1)^2 contains two or more distinct nonsquare perfect powers A097054.

Original entry on oeis.org

5, 11, 46, 2536, 558640, 572783, 3362407, 7928108, 8928803, 67460050, 106938971, 1763350849, 2501641555, 2756149047, 4584349318, 5713606932, 17941228664, 375376083513, 411124334926, 452894760105, 1167680330892, 1933159894790, 1946131548918, 2506032014606, 2507269866902, 8217688694093

Offset: 1

Hugo Pfoertner, Jul 21 2004

nonn

Empirically, there seem to be no intervals between consecutive squares containing more than two nonsquare perfect powers.

It is easy to see that two distinct powers between n^2 and (n+1)^2 are necessarily of the form x^p and y^q where p, q are distinct odd primes. Among the first 180 terms, only 4 are of type (p,q) = (3,7) and all others are of type (3,5). The first term with q = 11, if it exists, is > (1e6)^(11/2) = 1e33. - M. F. Hasler, Jan 18 2021

			a(1) = 5: 5^2 < 3^3 < 2^5 < 6^2,
a(2) = 11: 11^2 < 5^3 < 2^7 < 12^2,
a(3) = 46: 46^2 = 2116 < 3^7 = 2187 < 13^3 = 2197 < 47^2 = 2209.
a(4) = 2536: 2536^2 = 6431296 < 186^3 = 6434856 < 23^5 = 6436343 < 2537^2 = 6436369.
22 is not in the sequence because 2^9 and 8^3 (22^2 < 512 < 23^2) are not distinct.
Also, 181 is not listed since between 181^2 and 182^2 there is only 32^3 = 8^5.

T. D. Noe, Table of n, a(n) for n = 1..180 (using the b-file from A117934)

Cf. A000290, A097054, A097055.

Cf. A173341 (q=5), A173342 (q=7): y with a(n)^2 < y^q < (a(n)+1)^2.

is(n)=my(s,t); forprime(p=3,2*log(n+1.5)\log(2), t=floor((n+1)^(2/p)); if(t^p>n^2 && !ispower(t) && s++ > 1, return(1))); 0 \\ Charles R Greathouse IV, Dec 11 2012

haspow(lower,upper,eMin,eMax)=if(sqrtnint(upper,3)^3>lower, return(1)); forprime(e=eMin,eMax, if(sqrtnint(upper,e)^e>lower, return(1))); 0
list(lim)=lim\=1; my(v=List(),M=(lim+1)^2,L=logint(M,2),s); forprime(e=5,L, forprime(p=2,sqrtnint(M,e), s=sqrtint(p^e); if(haspow(s^2,(s+1)^2-1,e+1,L) && s<=lim, listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Nov 05 2015

a(5)-a(20) from Don Reble

a(21)-a(26) from David Wasserman, Dec 17 2007

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

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A097054 Nonsquare perfect powers.

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A097056 Numbers n such that the interval n^2 < x < (n+1)^2 contains two or more distinct nonsquare perfect powers A097054.

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

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Keywords

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Examples

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Programs

Mathematica

PARI

Formula