A001597 -id:A001597 - cp's OEIS frontend

A053289 First differences of consecutive perfect powers (A001597).

3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, 35, 19, 18, 39, 41, 43, 28, 17, 47, 49, 51, 53, 55, 57, 59, 61, 39, 24, 65, 67, 69, 71, 35, 38, 75, 77, 79, 81, 47, 36, 85, 87, 89, 23, 68, 71, 10, 12, 95, 97, 99, 101, 103, 40, 65, 107, 109, 100

Offset: 1

Labos Elemer, Mar 03 2000

nonn

Michel Waldschmidt writes: Conjecture 1.3 (Pillai). Let k be a positive integer. The equation x^p - y^q = k where the unknowns x, y, p and q take integer values, all >= 2, has only finitely many solutions (x,y,p,q). This means that in the increasing sequence of perfect powers [A001597] the difference between two consecutive terms [the present sequence] tends to infinity. It is not even known whether for, say, k=2, Pillai's equation has only finitely many solutions. A related open question is whether the number 6 occurs as a difference between two perfect powers. See Sierpiński [1970], problem 238a, p. 116. - Jonathan Vos Post, Feb 18 2008

Are there are any adjacent equal terms? - Gus Wiseman, Oct 08 2024

			Consecutive perfect powers are A001597(14) = 121, A001597(13) = 100, so a(13) = 121 - 100 = 21.

Wacław Sierpiński, 250 problems in elementary number theory, Modern Analytic and Computational Methods in Science and Mathematics, No. 26, American Elsevier, Warsaw, 1970, pp. 21, 115-116.
S. S. Pillai, On the equation 2^x - 3^y = 2^X - 3^Y, Bull, Calcutta Math. Soc. 37 (1945) 15-20.

Daniel Forgues and T. D. Noe, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Gaps between consecutive perfect powers, International Mathematical Forum, Vol. 11, No. 9 (2016), pp. 429-437.
Holly Krieger and Brady Haran, Catalan's Conjecture, Numberphile video (2018).
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.

For non-perfect-powers (A007916) we have A375706.
The union is A023055.
For prime-powers (A000961 or A246655) we have A057820.
Sorted positions of first appearances are A376268, complement A376519.
For second differences we have A376559.
Ascending and descending points are A376560 and A376561.
A001597 lists perfect-powers.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Cf. A025475, A069623, A219551.
Cf. A007921, A036263, A045542, A052410, A053707, A174965, A336416, A375735, A375736, A375740, A376562.

Differences@ Select[Range@ 3200, # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &] (* Michael De Vlieger, Jun 30 2016, after Ant King at A001597 *)

from sympy import mobius, integer_nthroot
def A053289(n):
    if n==1: return 3
    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)+1 >= kmax:
        kmax <<= 1
    rmin, rmax = 1, kmax
    while True:
        kmid = kmax+kmin>>1
        if f(kmid)+1 < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    while True:
        rmid = rmax+rmin>>1
        if f(rmid) < rmid:
            rmax = rmid
        else:
            rmin = rmid
        if rmax-rmin <= 1:
            break
    return kmax-rmax # Chai Wah Wu, Aug 13 2024

a(n) = A001597(n+1) - A001597(n). - Jonathan Vos Post, Feb 18 2008
From Amiram Eldar, Jun 30 2023: (Start)
Formulas from Jakimczuk (2016):
Lim sup_{n->oo} a(n)/(2*n) = 1.
Lim inf_{n->oo} a(n)/(2*n)^(2/3 + eps) = 0. (End)
Can be obtained by inserting 0 between 3 and 6 in A375702 and then adding 1 to all terms. In particular, for n > 2, a(n+1) - 1 = A375702(n). - Gus Wiseman, Sep 14 2024

A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

Original entry on oeis.org

1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, -16, -1, 21, 2, 2, -15, -11, 30, 2, 2, 2, 2, 2, 2, 2, -22, -15, 41, 2, 2, 2, -36, 3, 37, 2, 2, 2, -34, -11, 49, 2, 2, -66, 45, 3, -61, 2, 83, 2, 2, 2, 2, -63, 25, 42, 2, -9, -89

Offset: 1

Gus Wiseman, Sep 28 2024

sign

Perfect-powers A007916 are numbers with a proper integer root.

Does this sequence contain zero?

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we have A053289, union A023055, firsts A376268, A376519.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
For perfect-powers: A053289 (first differences), A376560 (positive curvature), A376561 (negative curvature).
For second differences: A036263 (prime), A073445 (composite), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A045542, A052410, A053707, A064113, A069623, A174965, A216765, A251092, A333254, A336416, A361102.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Differences[Select[Range[1000],perpowQ],2]

lista(nn) = my(v = concat (1, select(ispower, [1..nn])), w = vector(#v-1, i, v[i+1] - v[i])); vector(#w-1, i, w[i+1] - w[i]); \\ Michel Marcus, Oct 02 2024

from sympy import mobius, integer_nthroot
def A376559(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-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = bisection(f,n,n)
    b = bisection(lambda x:f(x)+1,a,a)
    return a+bisection(lambda x:f(x)+2,b,b)-(b<<1) # Chai Wah Wu, Oct 02 2024

A294068 Number of factorizations of n using perfect powers (elements of A001597) other than 1.

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, 2, 0, 0, 0, 2, 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, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, May 05 2018

nonn

			The a(1152) = 7 factorizations are (4*4*8*9), (4*8*36), (4*9*32), (8*9*16), (8*144), (9*128), (32*36).

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

Positions of zeros are A052485.

Cf. A000961, A001055, A001222, A001597, A001694, A007716, A007916, A045778, A052409, A052410, A052486, A091050, A203025, A303707, A303710.

ispp:= proc(n) local F;
  F:= ifactors(n)[2];
  igcd(op(map(t -> t[2],F)))>1
end proc:
f:= proc(n) local F, np, Q;
  F:= map(t -> t[2], ifactors(n)[2]);
  np:= mul(ithprime(i)^F[i],i=1..nops(F));
  Q:= select(ispp, numtheory:-divisors(np));
  G(Q,np)
end proc:
G:= proc(Q,n) option remember; local q,t,k;
    if not numtheory:-factorset(n) subset `union`(seq(numtheory:-factorset(q),q=Q)) then return 0 fi;
    q:= Q[1]; t:= 0;
    for k from 0 while n mod q^k = 0 do
      t:= t + procname(Q[2..-1],n/q^k)
    od;
    t
end proc:
G({},1):= 1:
map(f, [$1..200]); # Robert Israel, May 06 2018

ppQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]>1];
facsp[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsp[n/d],Min@@#>=d&]],{d,Select[Divisors[n],ppQ]}]];
Table[Length[facsp[n]],{n,100}]

A075802 Characteristic function of perfect powers, A001597.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 13 2002

nonn

Not multiplicative: for example, a(8)=a(9)=1, but a(72)=0. - Franklin T. Adams-Watters, Sep 09 2005

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.
Index entries for characteristic functions

Cf. A001597, A007916, A052409, A057427, A072292, A112526.

a075802 1 = 1
a075802 n = signum $ a052409 n - 1  -- Reinhard Zumkeller, May 26 2012

a[n_] := Boole[GCD @@ FactorInteger[n][[All, 2]] > 1]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 12 2011 *)

from sympy import perfect_power
def A075802(n): return int(bool(perfect_power(n))) if n>1 else 1 # Chai Wah Wu, Mar 11 2025

a(n) = A057427(A052409(n) - 1);

a(A001597(n))=1 and a(A007916(n))=0.

A025478 Least roots of perfect powers (A001597).

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 3, 2, 6, 7, 2, 3, 10, 11, 5, 2, 12, 13, 14, 6, 15, 3, 2, 17, 18, 7, 19, 20, 21, 22, 2, 23, 24, 5, 26, 3, 28, 29, 30, 31, 10, 2, 33, 34, 35, 6, 11, 37, 38, 39, 40, 41, 12, 42, 43, 44, 45, 2, 46, 3, 13, 47, 48, 7, 50, 51, 52, 14, 53, 54, 55, 5, 56, 57, 58, 15, 59, 60, 61, 62

Offset: 1

David W. Wilson

			a(5) = 2 because pp(5) = 16 = 2^4 (not 4^2 as we take the smallest base).

Daniel Forgues, Table of n, a(n) for n=1..10000

Cf. A052410 (least root), A001597 (perfect powers).

Cf. A025479 (largest exponents of perfect powers), A070228.

a025478 n = a025478_list !! (n-1)  -- a025478_list defined in A001597.
-- Reinhard Zumkeller, Mar 11 2014

pp = Select[ Range[5000], Apply[GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 &]; f[n_] := Block[{b = 2}, While[ !IntegerQ[ Log[b, pp[[n]]]], b++ ]; b]; Join[{1}, Table[ f[n], {n, 2, 80}]]
(* Second program: *)
Prepend[DeleteCases[#, 0], 1] &@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, Power[n, 1/e], 0] &@ FactorInteger@ n, {n, 4000}]  (* Michael De Vlieger, Apr 25 2017 *)

lista(kmax) = {my(r, e); print1(1, ", "); for(k = 1, kmax, e = ispower(k, , &r); if(e > 0, print1(r, ", ")));} \\ Amiram Eldar, Sep 07 2024

from math import gcd
from sympy import mobius, integer_nthroot, factorint
def A025478(n):
    if n == 1: return 1
    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 integer_nthroot(kmax, gcd(*factorint(kmax).values()))[0] # Chai Wah Wu, Aug 13 2024

a(n) = A052410(A001597(n)).

(i) a(n) < n for n > 2. (ii) a(n)/n is bounded and lim sup a(n)/n must be around 0.7. (iii) Sum_{k=1..n} a(k) seems to be asymptotic to c*n^2 with c around 0.29. (iv) a(n) = 2 if n is in A070228 (proof seems self-evident), hence there is no asymptotic expression for a(n) (just the average in (iii)). - Benoit Cloitre, Oct 14 2002

Definition edited and cross-reference added by Daniel Forgues, Mar 10 2009

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

A070428 Number of perfect powers (A001597) not exceeding 10^n.

Original entry on oeis.org

1, 4, 13, 41, 125, 367, 1111, 3395, 10491, 32670, 102231, 320990, 1010196, 3184138, 10046921, 31723592, 100216745, 316694005, 1001003332, 3164437425, 10004650118, 31632790244, 100021566157, 316274216762, 1000100055684

Offset: 0

Donald S. McDonald, May 15 2002

In the programs for this sequence, 4*n can be replaced by the smaller floor(n*log(10)/log(2)). - T. D. Noe, Nov 17 2006

			a(1) = 4 because the powers 1, 4, 8, 9 do not exceed 10^1.
a(2) = 13 because 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81 & 100, are the only perfect power numbers less than or equal to 100.

The Dominion (Wellington, NZ), 'wtd sell', 9 Nov. 1991.
sci.math, powers not exceeding n. nz science monthly advt, March 1993, 1:80 integers 1..10000 is perfect square or higher power.

Chai Wah Wu, Table of n, a(n) for n = 0..1999 (terms 0..999 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597.

Cf. A089579, A089580 (number of perfect powers (not including 1) < 10^n).

f[n_] := 1 - Sum[ MoebiusMu[x]*Floor[10^(n/x) - 1], {x, 2, n*Log2[10]}]; Array[f, 25, 0] (* Robert G. Wilson v, May 22 2009; modified Aug 04 2014 *)

for(n=0, 25, print1(sum(x=2, 4*n,-moebius(x)*(floor(10^(n/x)-1)))+1, ", ")); \\ Slightly modified by Jinyuan Wang, Mar 02 2020

from sympy import mobius, integer_nthroot
def A070428(n): return int(1-sum(mobius(x)*(integer_nthroot(10**n,x)[0]-1) for x in range(2,(10**n).bit_length()))) # Chai Wah Wu, Aug 13 2024

a(n) ~ sqrt(10^n).

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2002

Edited and extended by Robert G. Wilson v, Oct 11 2002

A025479 Largest exponents of perfect powers (A001597).

Original entry on oeis.org

2, 2, 3, 2, 4, 2, 3, 5, 2, 2, 6, 4, 2, 2, 3, 7, 2, 2, 2, 3, 2, 5, 8, 2, 2, 3, 2, 2, 2, 2, 9, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 10, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 11, 2, 7, 3, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 12, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 8, 2, 3, 2, 2, 2

Offset: 1

David W. Wilson

Greatest common divisor of all prime-exponents in canonical factorization of n-th perfect power. - Reinhard Zumkeller, Oct 13 2002

Asymptotically, 100% of the terms are 2, since the density of cubes and higher powers among the squares and higher powers is 0. - Daniel Forgues, Jul 22 2014

Daniel Forgues, Table of n, a(n) for n=1..10000

Cf. A001597, A025478, A052409, A124010, A322969.

a025479 n = a025479_list !! (n-1)  -- a025479_list is defined in A001597.
-- Reinhard Zumkeller, Mar 28 2014, Jul 15 2012

N:= 10^6: # to get terms corresponding to all perfect powers <= N
V:= Vector(N,storage=sparse);
V[1]:= 2:
for p from 2 to ilog2(N) do
  V[[seq(i^p,i=2..floor(N^(1/p)))]]:= p
od:
r,c,A := ArrayTools:-SearchArray(V):
convert(A,list); # Robert Israel, Apr 25 2017

Prepend[DeleteCases[#, 0], 2] &@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, e, 0] &@ FactorInteger@ n, {n, 10^4}] (* Michael De Vlieger, Apr 25 2017 *)

print1(2,", "); for(k=2, 3^8, if(j=ispower(k),print1(j,", "))) \\ Hugo Pfoertner, Jan 01 2019

a(n) = A052409(A001597(n)). - Reinhard Zumkeller, Oct 13 2002

A001597(n) = A025478(n)^a(n). - Reinhard Zumkeller, Mar 28 2014

Definition corrected by Daniel Forgues, Mar 07 2009

A023057 (Apparently) not the difference between adjacent perfect powers (A001597, integers of form a^b, a >= 1, b >= 2).

Original entry on oeis.org

6, 14, 22, 29, 31, 34, 42, 44, 46, 50, 52, 54, 58, 62, 64, 66, 70, 72, 78, 82, 84, 86, 88, 90, 91, 96, 98, 102, 105, 110, 111, 114, 117, 118, 120, 122, 124, 126, 130, 132, 134, 136, 140, 142, 153, 156, 158, 160, 162, 164, 165, 172, 176, 177, 178, 179, 181, 182, 188, 190

Offset: 1

David W. Wilson

Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Robert G. Wilson v, Table of n, a(n) for n = 1..4479
Alf van der Poorten, Remarks on the sequence of 'perfect' powers.

Cf. A001597 (perfect powers), A023055 (complement). See also A074980, A074981, A077286.

pp = Union[ Join[{1}, Flatten[ Table[n^i, {n, 2, Sqrt[10^12]}, {i, 2, Log[n, 10^12]}]]]]; l = Length[pp]; d = Sort[Take[pp, -l + 1] - Take[pp, l - 1]]; Complement[ Table[i, {i, 1, 200}], Take[ Union[d], 200]] (* Robert G. Wilson v *)

A175064 a(1) = 1; for n >= 2, a(n) = number of ways h to write the n-th perfect power A001597(n) as m^k with m >= 2 and k >= 1.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2

Offset: 1

Jaroslav Krizek, Jan 23 2010

nonn

Perfect powers with first occurrence of h >= 2: 4, 16, 64, 65536, 4096, ... [The perfect power corresponding to h is A175065(h) = 2^A005179(h). - Jianing Song, Oct 27 2024]

			For n = 11: A001597(11) = 64; there are 4 ways to write 64 as m^k: 64^1 = 8^2 = 4^3 = 2^6.

Cf. A253641, A253642, A000005, A001597.

from math import gcd
from sympy import mobius, integer_nthroot, divisor_count, factorint
def A175064(n):
    if n == 1: return 1
    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 divisor_count(gcd(*factorint(kmax).values())) # Chai Wah Wu, Aug 13 2024

a(n) = A000005(A253641(A001597(n))) = A253642(n)+1. - M. F. Hasler, Jan 25 2015

Extended by T. D. Noe, Apr 21 2011

Definition clarified by Jonathan Sondow, Nov 30 2012

cp's OEIS Frontend

A053289 First differences of consecutive perfect powers (A001597).

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A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

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A294068 Number of factorizations of n using perfect powers (elements of A001597) other than 1.

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A075802 Characteristic function of perfect powers, A001597.

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A025478 Least roots of perfect powers (A001597).

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

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A070428 Number of perfect powers (A001597) not exceeding 10^n.

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