A117453 -id:A117453 - cp's OEIS frontend

A175066 a(1) = 1, for n >= 2: a(n) = number of ways h to write perfect powers A117453(n) as m^k (m >= 2, k >= 2).

1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 2, 3, 4, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 3, 2, 5, 2, 2, 2, 2, 3, 5, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 7, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

Perfect powers with first occurrence of h >= 2: 16, 64, 65536, 4096, ... (A175065)
a(n) for n>1 is the subsequence of A253642 formed by the terms which exceed 1; equivalently, a(n)+1 for n>1 is the subsequence of A175064 formed by the terms which exceed 2. Also, sum of a(n)-1 over such n that A117453(n)<10^m gives A275358(m). - Andrey Zabolotskiy, Aug 16 2016
Numbers n such that a(n) is nonprime are 1, 26, 110, ... - Altug Alkan, Aug 22 2016

			For n = 12, A117453(12) = 4096 and a(12)=5 since there are 5 ways to write 4096 as m^k: 64^2 = 16^3 = 8^4 = 4^6 = 2^12.
729=27^2=9^3=3^6 and 1024=32^2=4^5=2^10 yield a(8)=a(9)=3. - _R. J. Mathar_, Jan 24 2010

Cf. A117453.

lista(nn) = {print1(1, ", "); for (i=2, nn, if (po = ispower(i), np = sum(j=2, po, ispower(i, j)); if (np>1, print1(np, ", "));););} \\ Michel Marcus, Mar 20 2013

from math import gcd
from sympy import mobius, integer_nthroot, factorint, divisor_count, primerange
def A175066(n):
    if n == 1: return 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): return int(n+sum(mobius(k)*(integer_nthroot(x,k)[0]-1+sum(integer_nthroot(x,p*k)[0]-1 for p in primerange((x//k).bit_length()))) for k in range(1,x.bit_length())))
    return divisor_count(gcd(*factorint(bisection(f,n,n)).values()))-1 # Chai Wah Wu, Nov 24 2024

If A117453(n) = m^k with k maximal, then a(n) = tau(k) - 1. - Charlie Neder, Mar 02 2019

Corrected and extended by R. J. Mathar, Jan 24 2010

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

A089580 Total number of perfect powers > 1 below 10^n, counting multiple representations separately.

Original entry on oeis.org

3, 15, 49, 143, 406, 1174, 3507, 10674, 32965, 102716, 321797, 1011533, 3186389, 10050743, 31730134, 100228040, 316713623, 1001037546, 3164497349, 10004755374, 31632975598, 100021893194, 316274794666, 1000101078148, 3162495003352, 10000467510247, 31623782520064, 100002164895587

Offset: 1

Martin Renner, Dec 29 2003

nonn

From Robert G. Wilson v, Jul 17 2016: (Start)
a(n) ~ sqrt(10^n).
a(n) - A089579(n) = A275358(n).
The four terms which make up the difference a(2) - A089579(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)
This sequence correlates (see Link) to A006880 via a power fit A*x^B. For example, using a(23) through a(29) one obtains (A,B) = (0.047272, 1.96592) with R^2 > 0.999999. This extrapolates A006880(30) as 1.46*10^28. The exponent well may be resolving to 2. - Bill McEachen, Mar 04 2025

			16 = 2^4 = 4^2 counts double, 256 = 2^8 = 4^4 = 16^2 counts three times.

Chai Wah Wu, Table of n, a(n) for n = 1..1998 (n = 1..100 from Robert G. Wilson, n = 101..400 from Karl-Heinz Hofmann)
Karl-Heinz Hofmann, Python program.
Bill McEachen, Plot A089580 vs A006880.

Cf. A001597, A072103, A117453, A275358, A060298.

Cf. A089579 (counting multiple representations only once).

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

Python
```
# see link.
```

a(n) = Sum_{k = 1..n} A060298(k). - Karl-Heinz Hofmann, Sep 18 2023

2 more terms from Martin Renner, Oct 02 2004
More terms from T. D. Noe, Nov 16 2006
More precise name by Hugo Pfoertner, Sep 16 2023

A089361 Numbers of pairs (i, j), i, j > 1, such that i^j <= n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15

Offset: 1

Cino Hilliard, Dec 27 2003

These numbers are related to the divergent series r sum(n^(1/k)) = n^(1/2) + n^(1/3) + ... + n^(1/r) for abs(n) > 0 and r=sqrt(n). Notice some numbers are missing, such as 4, 11, 12, 14.

Gaps (i.e., a(n) - a(n-1) > 1) occur for values of n > 1 in A117453. a(n) - a(n-1) = number of factors of j > 1, for the j in the pair (i,j) with the smallest value of i. Where n = A117453(x), a(n) = a(n-1) + A175066(x). For example: n = 64, a(64) = 13, a(63) = 10, 13 - 10 = 3; 64 = 2^6, 6 has three factors (2,3,6), corresponding to the three perfect powers for 64 (2^6, 4^3, 8^2). Also, A117453(3) = 64 and A175066(3) = 3. - Doug Bell, Jun 23 2015

			There are 5 perfect powers greater than 1 that are less than or equal to 16: 2^2, 2^3, 2^4, 3^2, 4^2, ergo the first 5 in the table.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001597, A117453, A175066, A259362, A381042.

N:= 1000; # to get a(1) to a(N)
B:= Vector(N);
for i from 2 to floor(sqrt(N)) do
  for j from 2 while i^j <= N do
    B[i^j]:= B[i^j]+1
  od
od:
convert(map(round,Statistics:-CumulativeSum(B)),list); # Robert Israel, Jun 24 2015

A089361[n_] := Sum[Floor[n^(1/j)] - 1, {j, 2, BitLength[n] - 1}];
Array[A089361, 100] (* Paolo Xausa, Jan 14 2025 *)

plessn(n,m=2) = { for(k=1,n, s=0; rx = sqrtint(k); ry = logint(k,2); for(x=m,rx, for(y=2,ry, p = floor(x^y); if(p<=k,s++) ) ); print1(s", ") ) } \\ [corrected by Jason Yuen, Jan 12 2025]

A = vector(100); for (p = 2, 6, i = 2; while (i^p <= 100, A[i^p]++; i++)); for (n = 2, 100, A[n] += A[n - 1]); \\ David Wasserman

a(n) = sum(j=2, logint(n,2), sqrtnint(n,j)-1) \\ Jason Yuen, Jan 12 2025

from sympy import integer_nthroot
def A089361(n): return sum(integer_nthroot(n,k)[0]-1 for k in range(2,n.bit_length())) # Chai Wah Wu, Nov 25 2024

a(1) = 0; for n > 1, if n not in A001597, a(n) = a(n-1), otherwise a(n) = a(n-1) + number of factors of j > 1 (A000005(j) - 1), for the j in the positive integer pair (i,j) where i^j = n with the smallest value of i. - Doug Bell, Jun 23 2015
a(n) = Sum_{j=2..floor(log_2(n))} floor(n^(1/j) - 1). - Robert Israel, Jun 24 2015
From Friedjof Tellkamp, Jun 14 2025: (Start)
a(n) = Sum_{k>=2..n} A259362(k), for n > 1.
G.f.: Sum_{j>=2, k>=2} x^(j^k)/(1-x). (End)

A322449 Numbers whose prime factorization contains only composite exponents.

Original entry on oeis.org

1, 16, 64, 81, 256, 512, 625, 729, 1024, 1296, 2401, 4096, 5184, 6561, 10000, 11664, 14641, 15625, 16384, 19683, 20736, 28561, 32768, 38416, 40000, 41472, 46656, 50625, 59049, 65536, 82944, 83521, 104976, 117649, 130321, 153664, 160000, 186624, 194481, 234256

Offset: 1

Alois P. Heinz, Dec 08 2018

nonn

Differs from A117453 first at n = 13: a(13) = 5184 = 2^6 * 3^4, A117453(13) = 6561 = 3^8.

			5184 = 2^6 * 3^4 is a term because all exponents are composite numbers.
1 is a term, because it has no prime factorization, and "the empty set has every property". - _N. J. A. Sloane_, Aug 25 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002808, A117453, A322448.

Join[{1},Select[Range[250000],AllTrue[FactorInteger[#][[;;,2]],CompositeQ]&]] (* Harvey P. Dale, Aug 25 2024 *)

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k in A002808} 1/p^k) = 1.1028952548... . - Amiram Eldar, Jul 02 2022

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

A340588 Squares of perfect powers.

Original entry on oeis.org

1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 6561, 10000, 14641, 15625, 16384, 20736, 28561, 38416, 46656, 50625, 59049, 65536, 83521, 104976, 117649, 130321, 160000, 194481, 234256, 262144, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1000000

Offset: 1

Terry D. Grant, Sep 21 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001597, A062965, A072102, A117453, A131605.

Cf. A153158 (complement within positive squares).

q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))<>2):
select(q, [i^2$i=1..1000])[];  # Alois P. Heinz, Nov 26 2024

Join[{1}, (Select[Range[2000], GCD @@ FactorInteger[#][[All, 2]] > 1 &])^2]

from sympy import mobius, integer_nthroot
def A340588(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**2 # Chai Wah Wu, Aug 14 2024

a(n) = A001597(n)^2.
a(n+1) = A062965(n) + 1. - Hugo Pfoertner, Sep 29 2020
Sum_{k>1} 1/(a(k) - 1) = 7/4 - Pi^2/6 = 7/4 - zeta(2).
Sum_{k>1} 1/a(k) = Sum_{k>=2} mu(k)*(1-zeta(2*k)).

A276108 Numbers expressible as perfect powers in a composite number of ways.

Original entry on oeis.org

1, 65536, 43046721, 68719476736, 152587890625, 2821109907456, 33232930569601, 281474976710656, 10000000000000000, 45949729863572161, 150094635296999121, 184884258895036416, 665416609183179841, 2177953337809371136, 6568408355712890625, 18446744073709551616

Offset: 1

Altug Alkan, Aug 27 2016

nonn

Old title was "Values of A117453(n) such that A175066(n) is not a prime number."
Terms are 1, 2^16, 3^16, 2^36, ...
Numbers m^k, where m is not a perfect power and k is a composite number in A154893 or 0. - Charlie Neder, Mar 02 2019

			65536 = 2^16 is a term because there are 4 corresponding ways that are 2^16, 4^8, 16^4, 256^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A117453, A154893, A175066.

from sympy import mobius, integer_nthroot, isprime, divisor_count
def A276108(n):
    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): return int(n+sum(mobius(k)*(integer_nthroot(x,k)[0]-1+sum(integer_nthroot(x,i*k)[0]-1 for i in range(2,(x//k).bit_length()) if isprime(i) or isprime(divisor_count(i)-1))) for k in range(1,x.bit_length())))
    return bisection(f,n,n) # Chai Wah Wu, Nov 25 2024

New title from Charlie Neder, Mar 04 2019

a(5)-a(16) from Chai Wah Wu, Nov 25 2024

A128993 A table which contains in each row two or more perfect powers with the same digits.

Original entry on oeis.org

125, 512, 144, 441, 169, 196, 961, 243, 324, 256, 625, 1024, 2401, 1089, 9801, 1296, 2916, 9216, 1369, 1936, 1728, 2187, 1764, 4761, 2197, 7921, 4096, 9604, 10201, 12100, 10404, 14400, 40401, 44100, 10609, 16900, 19600, 61009, 90601, 96100

Offset: 1

J. M. Bergot, Apr 30 2007

Perfect powers, A001597, may have anagrams (obtained by permutation of the digits, excluding anagrams with leading zeros) which are again perfect powers. Each row of the table collects a set of at least two different anagrams which are perfect powers.
Requiring at least two different representations in a row means that numbers like 81 = 3^4 = 9^2, which are in A117453, do not necessarily populate a row on their own.
The table is sorted such that entries in the first column are increasing, and such that each perfect power appears at most once.

			The table starts with the first 11 rows as follows:
125,512; 125=5^3 and 512 = 2^9 = 8^3
144,441; 144=12^2 and 441=21^2
169,196,961; 169=13^2 and 196=14^2 and 961=31^2
243,324; 243=3^5 and 324=18^2
256,625; 256 = 16^2=4^4 and 625=25^2=5^4
1024,2401; 1024=2^10=32^2 and 2401=49^2=7^4
1089,9801; 1089=33^2 and 9801=99^2
1296,2916,9216; 1296=36^2 and 2916=54^2 and 9216=96^2
1369,1936; 1369=37^2 and 1936=44^2
1728,2187; 1728=12^3 and 2187=3^7
1764,4761; 1764=42^2 and 4761=69^2

Cf. A117453.

Edited, and most terms replaced by R. J. Mathar, Nov 02 2009

cp's OEIS Frontend

A175066 a(1) = 1, for n >= 2: a(n) = number of ways h to write perfect powers A117453(n) as m^k (m >= 2, k >= 2).

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PARI

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

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SageMath

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A089580 Total number of perfect powers > 1 below 10^n, counting multiple representations separately.

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A089361 Numbers of pairs (i, j), i, j > 1, such that i^j <= n.

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Maple

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PARI

PARI

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A322449 Numbers whose prime factorization contains only composite exponents.

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

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