A097054 -id:A097054 - cp's OEIS frontend

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

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

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

A001597 Perfect powers: m^k where m > 0 and k >= 2.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764

Offset: 1

N. J. A. Sloane

Might also be called the nontrivial powers. - N. J. A. Sloane, Mar 24 2018
See A175064 for number of ways to write a(n) as m^k (m >= 1, k >= 1). - Jaroslav Krizek, Jan 23 2010
a(1) = 1, for n >= 2: a(n) = numbers m such that sum of perfect divisors of x = m has no solution. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) for n >= 2 is complement of A175082. - Jaroslav Krizek, Jan 24 2010
A075802(a(n)) = 1. - Reinhard Zumkeller, Jun 20 2011
Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.
For a proof of Catalan's conjecture, see the paper by Metsänkylä. - L. Edson Jeffery, Nov 29 2013
m^k is the largest number n such that (n^k-m)/(n-m) is an integer (for k > 1 and m > 1). - Derek Orr, May 22 2014
From Daniel Forgues, Jul 22 2014: (Start)
a(n) is asymptotic to n^2, since the density of cubes and higher powers among the squares and higher powers is 0. E.g.,
  a(10^1) = 49 (49% of 10^2),
  a(10^2) = 6400 (64% of 10^4),
  a(10^3) = 804357 (80.4% of 10^6),
  a(10^4) = 90706576 (90.7% of 10^8),
  a(10^n) ~ 10^(2n) - o(10^(2n)). (End)
A proper subset of A001694. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 6400, 804357, 90706576, 9565035601, 979846576384, 99066667994176, 9956760243243489, ... . - Robert G. Wilson v, Aug 15 2014

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D9.
René Schoof, Catalan's Conjecture, Springer-Verlag, 2008, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 1..10000.
Abdelkader Dendane, Power (Exponential) Calculator.
H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), p. 268, problem H-170.
Werner Hürlimann, A First Digit Theorem for Powers of Perfect Powers, Communications in Mathematics and Applications Vol. 5, No. 3 (2014), pp. 91-99.
Rafael Jakimczuk, On the Distribution of Perfect Powers, Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.5.
Rafael Jakimczuk, Asymptotic Formulae for the n-th Perfect Power, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.5.
Holly Krieger and Brady Haran, Catalan's Conjecture, Numberphile video (2018).
Tauno Metsänkylä, Catalan's conjecture: another old Diophantine problem solved, Bull. Amer. Math. Soc. (NS), Vol. 41, No. 1 (2004), pp. 43-57.
Preda Mihǎilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, Journal für die Reine und Angewandte Mathematik, vol. 27 (2004), pp. 167-195.
Donald J. Newman, A Problem Seminar, Springer; see Problem #72.
M. A. Nyblom, A Counting Function for the Sequence of Perfect Powers, Austral. Math. Soc. Gaz. 33 (2006), 338-343.
Hugo Pfoertner, 1010196 perfect powers up to 10^12, compressed 7z archive, 3.3 MB (2023).
Alf van der Poorten, Remarks on the sequence of 'perfect' powers.
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, 6th World Conference on 21st Century Mathematics 2013.
Eric Weisstein's World of Mathematics, Perfect Power.

Complement of A007916.
Subsequence of A072103; A072777 is a subsequence.
Union of A075109 and A075090.
Cf. A023055, A023057, A025478, A070428, A072102, A074981, A076404, A239728, A239870, A097054, A089579, A089580.
There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2), and which are sometimes confused with the present sequence.
First differences give A053289.

import Data.Map (singleton, findMin, deleteMin, insert)
a001597 n = a001597_list !! (n-1)
(a001597_list, a025478_list, a025479_list) =
   unzip3 $ (1, 1, 2) : f 9 (3, 2) (singleton 4 (2, 2)) where
   f zz (bz, ez) m
    | xx < zz = (xx, bx, ex) :
                f zz (bz, ez+1) (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    | xx > zz = (zz, bz, 2) :
                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, ex)) = findMin m  --  bx ^ ex == xx
-- Reinhard Zumkeller, Mar 28 2014, Oct 04 2012, Apr 13 2012

[1] cat [n : n in [2..1000] | IsPower(n) ];

isA001597 := proc(n)
    local e ;
    e := seq(op(2,p),p=ifactors(n)[2]) ;
    return ( igcd(e) >=2 or n =1 ) ;
end proc:
A001597 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA001597(a) then
                return a ;
            end if;
         end do;
    end if;
end proc:
seq(A001597(n),n=1..70) ; # R. J. Mathar, Jun 07 2011
N:= 10000: # to get all entries <= N
sort({1,seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))}); # Robert FERREOL, Jul 18 2023

min = 0; max = 10^4;  Union@ Flatten@ Table[ n^expo, {expo, Prime@ Range@ PrimePi@ Log2@ max}, {n, Floor[1 + min^(1/expo)], max^(1/expo)}] (* T. D. Noe, Apr 18 2011; slightly modified by Robert G. Wilson v, Aug 11 2014 *)
perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1765, perfectPowerQ] (* Ant King, Jun 29 2013; slightly modified by Robert G. Wilson v, Aug 11 2014 *)
nextPerfectPower[n_] := If[n == 1, 4, Min@ Table[ (Floor[n^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ n}]]; NestList[ nextPerfectPower, 1, 55] (* Robert G. Wilson v, Aug 11 2014 *)
Join[{1},Select[Range[2000],GCD@@FactorInteger[#][[All,2]]>1&]] (* Harvey P. Dale, Apr 30 2018 *)

{a(n) = local(m, c); if( n<2, n==1, c=1; m=1; while( cMichael Somos, Aug 05 2009 */

is(n)=ispower(n) || n==1 \\ Charles R Greathouse IV, Sep 16 2015

list(lim)=my(v=List(vector(sqrtint(lim\=1),n,n^2))); for(e=3,logint(lim,2), for(n=2,sqrtnint(lim,e), listput(v,n^e))); Set(v) \\ Charles R Greathouse IV, Dec 10 2019

from sympy import perfect_power
def ok(n): return n==1 or perfect_power(n)
print([m for m in range(1, 1765) if ok(m)]) # Michael S. Branicky, Jan 04 2021

import sympy
class A001597() :
    def _init_(self) :
        self.a = [1]
    def at(self, n):
        if n <= len(self.a):
            return self.a[n-1]
        else:
            cand = self.at(n-1)+1
            while sympy.perfect_power(cand) == False:
                cand += 1
            self.a.append(cand)
            return cand
a001597 = A001597()
for n in range(1,20):
    print(a001597.at(n)) # R. J. Mathar, Mar 28 2023

from sympy import mobius, integer_nthroot
def A001597(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,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 13 2024

def A001597_list(n) :
    return [k for k in (1..n) if k.is_perfect_power()]
A001597_list(1764) # Peter Luschny, Feb 03 2012

Goldbach showed that Sum_{n >= 2} 1/(a(n)-1) = 1.
Formulas from postings to the Number Theory List by various authors, 2002:
Sum_{i >= 2} Sum_{j >= 2} 1/i^j = 1;
Sum_{k >= 2} 1/(a(k)+1) = Pi^2 / 3 - 5/2;
Sum_{k >= 2} 1/a(k) = Sum_{n >= 2} mu(n)(1- zeta(n)) approx = 0.87446436840494... See A072102.
For asymptotics see Newman.
For n > 1: gcd(exponents in prime factorization of a(n)) > 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
a(n) ~ n^2. - Thomas Ordowski, Nov 04 2012
a(n) = n^2 - 2*n^(5/3) - 2*n^(7/5) + (13/3)*n^(4/3) - 2*n^(9/7) + 2*n^(6/5) - 2*n^(13/11) + o(n^(13/11)) (Jakimczuk, 2012). - Amiram Eldar, Jun 30 2023

Minor corrections from N. J. A. Sloane, Jun 27 2010

A070265 Odd powers: numbers n = m^e with e > 1 odd.

Original entry on oeis.org

1, 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, 729, 1000, 1024, 1331, 1728, 2048, 2187, 2197, 2744, 3125, 3375, 4096, 4913, 5832, 6859, 7776, 8000, 8192, 9261, 10648, 12167, 13824, 15625, 16384, 16807, 17576, 19683, 21952, 24389, 27000

Offset: 1

Eric W. Weisstein, May 07 2002

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Odd Powers.

Cf. A001597, A008683, A097054.

N:= 10^6: # to get all terms <= N
{1,seq(seq(a^(2*k+1), k = 1 .. floor((log[a](N)-1)/2)),a=2..floor(N^(1/3)))};
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Apr 24 2015

nn = 27000; Join[{1}, Union[Flatten[Table[n^i, {i, Prime[Range[2, PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]]] (* T. D. Noe, Apr 19 2011 *)

is(x)=p=ispower(x);x==1||(p>1&bitand(p,p-1)!=0) \\ Charles R Greathouse IV, Apr 20 2015; corrected by Jeppe Stig Nielsen, Jul 14 2015

list(lim)=my(v=List([1])); forstep(e=3,log(lim)\log(2),2, for(n=2,sqrtnint(lim\1,e), listput(v,n^e))); Set(v) \\ Charles R Greathouse IV, Apr 20 2015

from sympy import mobius, integer_nthroot
def A070265(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(3,x.bit_length(),2)))
    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

a(n) ~ n^3. - Charles R Greathouse IV, Apr 20 2015

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} mu(2*k+1)*(1-zeta(2*k+1)) = 1.2479294392... - Amiram Eldar, Dec 21 2020

Name clarified by Charles R Greathouse IV, Oct 16 2015

A239728 Perfect power but neither square nor cube.

Original entry on oeis.org

32, 128, 243, 2048, 2187, 3125, 7776, 8192, 16807, 78125, 100000, 131072, 161051, 177147, 248832, 279936, 371293, 524288, 537824, 759375, 823543, 1419857, 1594323, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 8388608, 10000000, 11881376, 17210368

Offset: 1

Jeppe Stig Nielsen, Mar 25 2014

			279936 is included since 279936 = 6^7 is a power and this is not a square or a cube.
59049 = 9^5 not included since this is a square 243^2 = 59049.
32768 = 8^5 not included since this is a cube 32^3 = 32768.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A001597 (perfect powers), A097054 (nonsquare perfect powers), A340585 (noncube perfect powers).

Cf. A008683, A052409.

for(i=1, 2^25, if(gcd(ispower(i), 6) == 1, print(i)))

from sympy import mobius, integer_nthroot
def A239728(n):
    def f(x): return int(n+x-integer_nthroot(x,5)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(7,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

GCD(A052409(a(n)), 6) = 1. - Reinhard Zumkeller, Mar 28 2014

Sum_{n>=1} 1/a(n) = 1 - zeta(2) - zeta(3) + zeta(6) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 0.0448164603... - Amiram Eldar, Dec 21 2020

A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

Original entry on oeis.org

4, 9, 25, 225, 676, 2116, 6724, 7921, 8100, 16641, 104329, 131044, 160801, 176400, 372100, 389376, 705600, 4096576, 7306209, 7884864, 47444544, 146385801, 254817369, 373262400, 607622500, 895804900, 1121580100, 1330936324, 1536875209, 2097182025, 2258435529, 2749953600

Offset: 1

Hugo Pfoertner, Jan 14 2021

nonn

Apparently, all known terms (checked through 10^18) are squares with maximum exponent 2, i.e., terms of A111245 (squares that are not a higher power). This would imply that of 3 immediately adjacent perfect powers, at least one is a term of A111245. Is there a known counterexample of 3 consecutive perfect powers, none of which is in A111245?

			The first terms, assuming 1 being at least a cube:
.
  n   p1  x^p1  p2  a(n)  p3  z^p3
                   =y^p2
  1  >2     1   2     4   3     8
  2   3     8   2     9   4    16
  3   4    16   2    25   3    27
  4   3   216   2   225   5   243
  5   4   625   2   676   6   729

Hugo Pfoertner, Table of n, a(n) for n = 1..181 (terms < 10^18)

Cf. A001597, A025479, A076467, A097054, A111245, A153158, A340643, A340700, A340701.

a340642(limit)={my(p2=999, p1=2, n2=1, n1=4); for(n=5, limit, my(p0=ispower(n)); if(p0>1, if(p2>2&p0>2, print1(n1,", ")); n2=n1; n1=n; p2=p1; p1=p0))};
a340642(50000000)

A239870 Noncube perfect powers. [Warning: definition does not match the DATA.].

Original entry on oeis.org

4, 9, 16, 32, 36, 49, 81, 121, 128, 144, 169, 196, 243, 256, 324, 400, 441, 484, 576, 625, 841, 900, 961, 1024, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1849, 1936, 2025, 2048, 2187, 2209, 2304, 2401, 2601, 2704, 2916, 3025, 3125, 3249, 3364, 3600

Offset: 1

Reinhard Zumkeller, Mar 28 2014

The NAME suggests that this is an erroneous version of A340585 (which includes 25, for example), but the Haskell implementation indicates that the true definition is more complicated. - R. J. Mathar, Jan 13 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A097054, A239728, intersection of A007412 and A001597.

import Data.Map (singleton, findMin, deleteMin, insert)
a239870 n = a239870_list !! (n-1)
a239870_list = f 9 (3, 2) (singleton 4 (2, 2)) where
   f zz (bz, ez) m
    | xx < zz = if ex `mod` 3 > 0
      then xx : f zz (bz, ez+1) (insert (bx*xx) (bx, ex+1) $ deleteMin m)
      else      f zz (bz, ez+1) (insert (bx*xx) (bx, ex+1) $ deleteMin m)
    | xx > zz = if ez `mod` 3 > 0
      then zz : f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)
      else      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, ex)) = findMin m  --  bx ^ ex == xx

A052409(a(n)) mod 3 > 0.

A340585 Noncube perfect powers.

Original entry on oeis.org

4, 9, 16, 25, 32, 36, 49, 81, 100, 121, 128, 144, 169, 196, 225, 243, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2048, 2116, 2187, 2209, 2304, 2401, 2500

Offset: 1

Hugo Pfoertner, Jan 12 2021

nonn

This was the original definition of A239870. However, the true definition of that sequence seems to be slightly different.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000578, A001597, A002117, A007412, A072102, A076467, A097054, A239728.

filter:= proc(n) local g;
  g:= igcd(op(ifactors(n)[2][..,2]));
  g > 1 and (g mod 3 <> 0)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jan 12 2021

Select[Range[2, 2500], (g = GCD @@ FactorInteger[#][[;; , 2]]) > 1 && !Divisible[g, 3] &] (* Amiram Eldar, Jan 12 2021 *)

for(n=2,2500,if( ispower(n) % 3, print1(n,", ")))

from math import isqrt
from sympy import mobius, integer_nthroot
def A340585(n):
    def f(x): return int(n+x-isqrt(x)+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(5,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

Sum_{n>=1} 1/a(n) = 1 - zeta(3) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 1 - A002117 + A072102 = 0.6724074652... - Amiram Eldar, Jan 12 2021

A384518 Nonsquarefree numbers that are squarefree numbers raised to an odd power.

Original entry on oeis.org

8, 27, 32, 125, 128, 216, 243, 343, 512, 1000, 1331, 2048, 2187, 2197, 2744, 3125, 3375, 4913, 6859, 7776, 8192, 9261, 10648, 12167, 16807, 17576, 19683, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653, 54872, 59319, 68921, 74088, 78125, 79507, 97336, 100000

Offset: 1

Amiram Eldar, Jun 01 2025

nonn

Subsequence of A097054 and first differs from it at n = 12: A097054(12) = 1728 = 2^6 * 3^3 is not a term of this sequence.

Numbers whose prime factorization exponents are equal, odd and larger than 1.

Intersection of A072777 and A268335.
Equals A072777 \ A384517.
Subsequence of A097054.
Cf. A005117.

Select[Range[10^5], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && OddQ[u[[1]]] &]

isok(k) = {my(s, e = ispower(k, , &s)); e % 2 && issquarefree(s);}

from math import isqrt
from sympy import mobius, integer_nthroot
def A384518(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1		
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(3,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Jun 01 2025

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k+1)/zeta(4*k+2)-1) = 0.22841193284408713846... .

cp's OEIS Frontend

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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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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Mathematica

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A001597 Perfect powers: m^k where m > 0 and k >= 2.

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A070265 Odd powers: numbers n = m^e with e > 1 odd.

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Maple

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A239728 Perfect power but neither square nor cube.

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A340642 Perfect powers such that the two immediately adjacent perfect powers both have an exponent greater than 2.

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