A070428 -id:A070428 - cp's OEIS frontend

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

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

A052486 Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.

Original entry on oeis.org

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6075

Offset: 1

Henry Bottomley, Mar 16 2000

nonn

Number of terms < 10^n: 0, 1, 13, 60, 252, 916, 3158, 10553, 34561, 111891, 359340, 1148195, 3656246, 11616582, 36851965, ..., A118896(n) - A070428(n). - Robert G. Wilson v, Aug 11 2014

a(n) = (s(n))^2 * f(n), s(n) > 1, f(n) > 1, where s(n) is not a power of f(n), and f(n) is squarefree and gcd(s(n), f(n)) = f(n). - Daniel Forgues, Aug 11 2015

			a(3)=200 because 200=2^3*5^2, both 3 and 2 are greater than 1, and the highest common factor of 3 and 2 is 1.
Factorizations of a(1) to a(20):
    72 = 2^3  3^2,  108 = 2^2 3^3,  200 = 2^3 5^2,  288 = 2^5  3^2,
   392 = 2^3  7^2,  432 = 2^4 3^3,  500 = 2^2 5^3,  648 = 2^3  3^4,
   675 = 3^3  5^2,  800 = 2^5 5^2,  864 = 2^5 3^3,  968 = 2^3 11^2,
   972 = 2^2  3^5, 1125 = 3^2 5^3, 1152 = 2^7 3^2, 1323 = 3^3  7^2,
  1352 = 2^3 13^2, 1372 = 2^2 7^3, 1568 = 2^5 7^2, 1800 = 2^3  3^2 5^2.
Examples for a(n) = (s(n))^2 * f(n): (see above comment)
s(n) = 6,  6, 10, 12, 14, 12, 10, 18, 15, 20, 12, 22, 18, 15, 24, 21,
f(n) = 2,  3,  2,  2,  2,  3,  5,  2,  3,  2,  6,  2,  3,  5,  2,  3,

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Project Euler, Problem 302: Strong Achilles Numbers.
Robert Israel, log-log plot of a(n).
Eric Weisstein's World of Mathematics, Achilles Number.
OEIS Wiki, Achilles numbers.

Cf. A001597, A001694, A007916, A072102, A082695.

filter:= proc(n) local E; E:= map(t->t[2], ifactors(n)[2]); min(E)>1 and igcd(op(E))=1 end proc:
select(filter,[$1..10000]); # Robert Israel, Aug 11 2014

achillesQ[n_] := Block[{ls = Last /@ FactorInteger@n}, Min@ ls > 1 == GCD @@ ls]; Select[ Range@ 5500, achillesQ@# &] (* Robert G. Wilson v, Jun 10 2010 *)

is(n)=my(f=factor(n)[,2]); n>9 && vecmin(f)>1 && gcd(f)==1 \\ Charles R Greathouse IV, Sep 18 2015, replacing code by M. F. Hasler, Sep 23 2010

from math import gcd
from itertools import count, islice
from sympy import factorint
def A052486_gen(startvalue=1): # generator of terms >= startvalue
    return (n for n in count(max(startvalue,1)) if (lambda x: all(e > 1 for e in x) and gcd(*x) == 1)(factorint(n).values()))
A052486_list = list(islice(A052486_gen(),20)) # Chai Wah Wu, Feb 19 2022

from math import isqrt
from sympy import mobius, integer_nthroot
def A052486(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x+1, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length()))
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

a(n) = O(n^2). - Daniel Forgues, Aug 11 2015
a(n) = O(n^2 / log log n). - Daniel Forgues, Aug 12 2015
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} mu(k)*(1-zeta(k)) - 1 = A082695 - A072102 - 1 = 0.06913206841581433836...  - Amiram Eldar, Oct 14 2020

Example edited by Mac Coombe (mac.coombe(AT)gmail.com), Sep 18 2010

Name edited by M. F. Hasler, Jul 17 2019

A188951 Number of perfect powers (A001597) < 2^n.

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 10, 15, 22, 30, 41, 57, 81, 113, 155, 216, 298, 416, 582, 813, 1135, 1588, 2223, 3115, 4368, 6135, 8622, 12127, 17063, 24022, 33838, 47688, 67226, 94804, 133737, 188709, 266350, 376018, 530940, 749819, 1059096, 1496143, 2113801, 2986769

Offset: 0

T. D. Noe, Apr 20 2011

nonn

			For n=3, the perfect powers smaller than 2^3=8 are: 1 and 4. So a(3) = 2.

Amiram Eldar, Table of n, a(n) for n = 0..6643

Cf. A001597, A070228, A070428 (perfect powers not exceeding 10^n).

Join[{0,1}, Table[-Sum[MoebiusMu[x]*Floor[2^(n/x) - 1], {x, 2, n}], {n, 2, 50}]]

a(n) = sum(k=1, 2^n-1, (k==1) || ispower(k)); \\ Michel Marcus, Apr 11 2016

from sympy import mobius, integer_nthroot
def A188951(n): return int(sum(mobius(x)*(1-integer_nthroot(1<Chai Wah Wu, Aug 13 2024

a(n) = A070228(n) - 1 for n > 1. - Amiram Eldar, May 19 2022

A075308 Number of n-digit perfect powers.

Original entry on oeis.org

4, 8, 28, 84, 242, 744, 2284, 7096, 22179, 69561, 218759, 689206, 2173942, 6862783, 21676671, 68493153, 216477260, 684309327, 2163434093, 6840212693, 21628140126, 68388775913, 216252650605, 683825838922, 2162393136881, 6837971108286, 21623312527390, 68378377967873

Offset: 1

Zak Seidov, Oct 11 2002

			a(2) = 8 because there are eight 2-digit perfect powers: 16, 25, 27, 32, 36, 49, 64, 81.
a(3) = 28 = A070428(3) - A070428(2) = 41 - 13 (in A070428 offset is 0).

Cf. A001597, A070428.

from sympy import mobius, integer_nthroot
def A075308(n): return int(sum(mobius(x)*(integer_nthroot(10**(n-1),x)[0]-integer_nthroot(10**n,x)[0]) for x in range(2,((10**(n-1)).bit_length())))-sum(mobius(x)*(integer_nthroot(10**n,x)[0]-1) for x in range((10**(n-1)).bit_length(),(10**n).bit_length()))) if n>2 else n<<2 # Chai Wah Wu, Aug 13 2024

a(n) = A070428(n) - A070428(n-1) for n >= 3.

More terms from Jinyuan Wang, Mar 02 2020

A089579 Total number of perfect powers > 1 below 10^n.

Original entry on oeis.org

3, 11, 39, 123, 365, 1109, 3393, 10489, 32668, 102229, 320988, 1010194, 3184136, 10046919, 31723590, 100216743, 316694003, 1001003330, 3164437423, 10004650116, 31632790242, 100021566155, 316274216760, 1000100055682, 3162493192563, 10000464300849, 31623776828239, 100002154796112

Offset: 1

Martin Renner, Dec 29 2003

nonn

k is a perfect power <=> there exist integers a and b, b > 1, and k = a^b.
From Robert G. Wilson v, Jul 17 2016: (Start)
Limit_{n->oo} a(n)/sqrt(10^n) = 1.
A089580(n) - a(n) = A275358(n).
The four terms which make up the difference between A089580(2) - a(2) are: 16 = 2^4 = 4^2, 64 = 2^6 = 4^3 = 8^2 and 81 = 3^4 = 9^2; one for 16, two for 64 and one for 81 making a total of 4. See A117453.
(End)

			For n=2, the 11 perfect powers > 1 below 10^2 = 100 are: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81. - _Michael B. Porter_, Jul 18 2016

Robert G. Wilson v, Table of n, a(n) for n = 1..100

Cf. A001597, A070428, A089580, A275358.

Table[lim=10^n-1; Sum[ -(Floor[lim^(1/k)]-1)*MoebiusMu[k], {k,2,Floor[Log[2,lim]]}], {n,30}] (* T. D. Noe, Nov 16 2006 *)

from sympy import mobius, integer_nthroot
def A089579(n): return int(sum(mobius(x)*(1-integer_nthroot(10**n,x)[0]) for x in range(2,(10**n).bit_length())))-1 if n>1 else 3 # Chai Wah Wu, Aug 13 2024

def A089579(n):
    gen = (p for p in srange(2, 10^n) if p.is_perfect_power())
    return sum(1 for _ in gen)
print([A089579(n) for n in range(1, 7)])  # Peter Luschny, Sep 15 2023

a(n) = A070428(n) - 2 for n >= 2.

a(9)-a(10) from Martin Renner, Oct 02 2004
More terms from T. D. Noe, Nov 16 2006
More precise name by Hugo Pfoertner, Sep 15 2023

A070228 Number of perfect powers (A001597) not exceeding 2^n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 16, 23, 31, 42, 58, 82, 114, 156, 217, 299, 417, 583, 814, 1136, 1589, 2224, 3116, 4369, 6136, 8623, 12128, 17064, 24023, 33839, 47689, 67227, 94805, 133738, 188710, 266351, 376019, 530941, 749820, 1059097, 1496144, 2113802, 2986770, 4220666

Offset: 0

Donald S. McDonald, May 14 2002

			How many powers are there not exceeding 2^4?: 1, 4, 8, 9, 16. Hence a(4) = 5.
a(22)=2224: there are 2224 powers not exceeding 2^22.

Amiram Eldar, Table of n, a(n) for n = 0..6643 (terms 0..1000 from Robert G. Wilson v)

Cf. A070428, A188951.

f[n_] := 1 - Sum[ MoebiusMu[x]*Floor[2^(n/x) - 1], {x, 2, n}]; Array[f, 44, 0] (* Robert G. Wilson v, Jan 20 2015 *)

a(n) = 1 - sum(k=2, n, moebius(k)*(sqrtnint(2^n,k)-1));

from sympy import mobius, integer_nthroot
def A070228(n): return int(1+sum(mobius(x)*(1-integer_nthroot(1<Chai Wah Wu, Aug 13 2024

a(n) = 1 - Sum_{k=2..n} Moebius(k)*floor(2^(n/k)-1). - Robert G. Wilson v, Jan 20 2015

a(n) = A188951(n) + 1 for n > 1. - Amiram Eldar, May 19 2022

a(39)-a(44) from Alex Ratushnyak, Jan 02 2014

A259362 a(1) = 1, for n > 1: a(n) is the number of ways to write n as a nontrivial perfect power.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Doug Bell, Jun 24 2015

nonn

a(n) = number of integer pairs (i,j) for distinct values of i where i > 0, j > 1 and n = i^j. Since 1 = 1^r for all real values of r, the requirement for a distinct i causes a(1) = 1 instead of a(1) = infinity.
Alternatively, the sequence can be defined as: a(1) = 1, for n > 1: a(n) = number of pairs (i,j) such that i > 0, j > 1 and n = i^j.
A007916 = n, where a(n) = 0.
A001597 = n, where a(n) > 0.
A175082 = n, where n = 1 or a(n) = 0.
A117453 = n, where n = 1 or a(n) > 1.
A175065 = n, where n > 1 and a(n) > 0 and this is the first occurrence in this sequence of a(n).
A072103 = n repeated a(n) times where n > 1.
A075802 = min(1, a(n)).
A175066 = a(n), where n = 1 or a(n) > 1. This sequence is an expansion of A175066.
A253642 = 0 followed by a(n), where n > 1 and a(n) > 0.
A175064 = a(1) followed by a(n) + 1, where n > 1 and a(n) > 0.
Where n > 1, A001597(x) = n (which implies a(n) > 0), i = A025478(x) and j = A253641(n), then a(n) = A000005(j) - 1, which is the number of factors of j greater than 1. The integer pair (i,j) comprises the smallest value i and the largest value j where i > 0, j > 1 and n = i^j. The a(n) pairs of (a,b) where a > 0, b > 1 and n = a^b are formed with b = each of the a(n) factors of j greater than 1. Examples for n = {8,4096}:
  a(8) = 1, A001597(3) = 8, A025478(3) = 2, A253641(8) = 3, 8 = 2^3 and A000005(3) - 1 = 1 because there is one factor of 3 greater than 1 [3]. The set of pairs (a,b) is {(2,3)}.
  a(4096) = 5, A001597(82) = 4096, A025478(82) = 2, A253641(4096) = 12, 4096 = 2^12 and A000005(12) - 1 = 5 because there are five factors of 12 greater than 1 [2,3,4,6,12]. The set of pairs (a,b) is {(64,2),(16,3),(8,4),(4,6),(2,12)}.
A023055 = the ordered list of x+1 with duplicates removed, where x is the number of consecutive zeros appearing in this sequence between any two nonzero terms.
A070428(x) = number of terms a(n) > 0 where n <= 10^x.
a(n) <= A188585(n).

			a(6) = 0 because there is no way to write 6 as a nontrivial perfect power.
a(9) = 1 because there is one way to write 9 as a nontrivial perfect power: 3^2.
a(16) = 2 because there are two ways to write 16 as a nontrivial perfect power: 2^4, 4^2.
From _Friedjof Tellkamp_, Jun 14 2025: (Start)
n:       1, 2, 3, 4, 5, 6, 7, 8, 9, ...
Squares: 1, 0, 0, 1, 0, 0, 0, 0, 1, ... (A010052)
Cubes:   1, 0, 0, 0, 0, 0, 0, 1, 0, ... (A010057)
...
Sum:    oo, 0, 0, 1, 0, 0, 0, 1, 1, ...
a(1)=1:  1, 0, 0, 1, 0, 0, 0, 1, 1, ... (= this sequence). (End)

Doug Bell, Table of n, a(n) for n = 1..5000

Cf. A001597, A007916, A023055, A025478, A070428, A072103, A075090, A075109, A075802, A117453, A175064, A175066, A175082, A188585, A253641, A253642.

Cf. A010052, A010057, A374016, A089361, A089723, A382691 (alternating).

a[n_] := If[n == 1, 1, Sum[Boole[IntegerQ[n^(1/k)]], {k, 2, Floor[Log[2, n]]}]]; Array[a, 100] (* Friedjof Tellkamp, Jun 14 2025 *)
a[n_] := If[n == 1, 1, DivisorSigma[0, Apply[GCD, Transpose[FactorInteger[n]][[2]]]] - 1]; Array[a, 100] (* Michael Shamos, Jul 06 2025 *)

a(n) = if (n==1, 1, sum(i=2, logint(n, 2), ispower(n, i))); \\ Michel Marcus, Apr 11 2025

a(1) = 1, for n > 1: a(n) = A000005(A253641(n)) - 1.
If n not in A001597, then a(n) = 0, otherwise a(n) = A175064(x) - 1 where A001597(x) = n.
From Friedjof Tellkamp, Jun 14 2025: (Start)
a(n) = A089723(n) - 1, for n > 1.
a(n) = A010052(n) + A010057(n) + A374016(n) + (...), for n > 1.
Sum_{k>=2..n} a(k) = A089361(n), for n > 1.
G.f.: x + Sum_{j>=2, k>=2} x^(j^k).
Dirichlet g.f.: 1 + Sum_{k>=2} zeta(k*s)-1. (End)

A275358 The difference between A089580(n) and A089579(n).

Original entry on oeis.org

0, 4, 10, 20, 41, 65, 114, 185, 297, 487, 809, 1339, 2253, 3824, 6544, 11297, 19620, 34216, 59926, 105258, 185356, 327039, 577906, 1022466, 1810789, 3209398, 5691825, 10099475, 17927609, 31833805, 56541947, 100449345, 178484340, 317187186, 563744378, 1002052726

Offset: 1

Robert G. Wilson v, Jul 24 2016

nonn

Submitted on the request of Omar E. Pol 17 July 2016. (A089579).

a(n) is the sum of A175066(m)-1 over such m that A117453(m)<10^n. - Andrey Zabolotskiy, Aug 17 2016

			a(2) = A089580(2)-A089579(2) = 4 because of the three terms: 16 = 2^4 = 4^2, 64 = 2^6 = 4^3 = 8^2 and 81 = 3^4 = 9^2; one for 16, two for 64 and one for 81 making a total of 4.

Robert G. Wilson v, Table of n, a(n) for n = 1..100

Cf. A011007, A011557, A070428, A089579, A089580.

f[n_] := Block[{lim = 10^n -1}, Sum[ (Floor[ lim^(1/k)] - 1)(1 + MoebiusMu[k]), {k, 2, Floor[ Log[2, lim]]}]]; Array[f, 36]

a(n) = A089580(n) - A089579(n).

Limit_{n->oo} a(n+1)/a(n) = 1.778279... (A011007). - Altug Alkan, Aug 22 2016

A362973 The number of cubefull numbers (A036966) not exceeding 10^n.

Original entry on oeis.org

1, 2, 7, 20, 51, 129, 307, 713, 1645, 3721, 8348, 18589, 41136, 90619, 198767, 434572, 947753, 2062437, 4480253, 9718457, 21055958, 45575049, 98566055, 213028539, 460160083, 993533517, 2144335391, 4626664451, 9980028172, 21523027285, 46408635232, 100053270534

Offset: 0

Amiram Eldar, May 11 2023

nonn

The number of cubefull numbers not exceeding x is N(x) = c_0 * x^(1/3) + c_1 * x^(1/4) + c_2 * x^(1/5) + o(x^(1/8)), where c_0 (A362974), c_1 (A362975) and c_2 (A362976) are constants (Bateman and Grosswald, 1958; Finch, 2003).

The digits of a(3k) converge to A362974 as k -> oo. - Chai Wah Wu, May 13 2023

			There are 2 cubefull numbers not exceeding 10, 1 and 8, therefore a(1) = 2.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.6.1, pp. 113-115.

Chai Wah Wu, Table of n, a(n) for n = 0..36
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois Journal of Mathematics, Vol. 2, No. 1 (1958), pp. 88-98.
A. Ivić and P. Shiu, The distribution of powerful integers, Illinois Journal of Mathematics, Vol. 26, No. 4 (1982), pp. 576-590.
Ekkehard Krätzel, On the distribution of square-full and cube-full numbers, Monatshefte für Mathematik, Vol. 120, No. 2 (1995), pp. 105-119.
P. Shiu, The distribution of cube-full numbers, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295.
P. Shiu, Cube-full numbers in short intervals, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 112, No. 1 (1992), pp. 1-5.

Cf. A036966, A362974, A362975, A362976.

Similar sequences: A070428, A118896.

a[n_] := Module[{max = 10^n}, CountDistinct@ Flatten@ Table[i^5 * j^4 * k^3, {i, Surd[max, 5]}, {j, Surd[max/i^5, 4]}, {k, CubeRoot[max/(i^5*j^4)]}]]; Array[a, 15, 0]

from math import gcd
from sympy import factorint, integer_nthroot
def A362973(n):
    m, c = 10**n, 0
    for x in range(1,integer_nthroot(m,5)[0]+1):
        if all(d<=1 for d in factorint(x).values()):
            for y in range(1,integer_nthroot(z:=m//x**5,4)[0]+1):
                if gcd(x,y)==1 and all(d<=1 for d in factorint(y).values()):
                    c += integer_nthroot(z//y**4,3)[0]
    return c # Chai Wah Wu, May 11-13 2023

a(23)-a(31) from Chai Wah Wu, May 11 2023

cp's OEIS Frontend

A077284 Duplicate of A070428.

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

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A052486 Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.

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A188951 Number of perfect powers (A001597) < 2^n.

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A075308 Number of n-digit perfect powers.

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A089579 Total number of perfect powers > 1 below 10^n.

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A070228 Number of perfect powers (A001597) not exceeding 2^n.