A075090 -id:A075090 - 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

A075109 Odd perfect powers (1 together with numbers m^k, m odd, k >= 2).

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 121, 125, 169, 225, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1331, 1369, 1521, 1681, 1849, 2025, 2187, 2197, 2209, 2401, 2601, 2809, 3025, 3125, 3249, 3375, 3481, 3721, 3969, 4225, 4489, 4761, 4913, 5041, 5329, 5625

Offset: 1

Zak Seidov, Oct 11 2002

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

Intersection of A001597 and A005408.

Cf. A008683, A075090.

a075109 n = a075109_list !! (n-1)
a075109_list = filter odd a001597_list  -- Reinhard Zumkeller, Oct 04 2012

[1] cat [n : n in [3..6000 by 2] | IsPower(n) ]; // Vincenzo Librandi, Mar 31 2014

q:= n-> n=1 or n::odd and igcd(seq(i[2], i=ifactors(n)[2]))>1:
select(q, [$1..6000])[];  # Alois P. Heinz, May 04 2022

Take[Union[Flatten[Table[a^b, {a, 1, 99, 2}, {b, 2, 15}]]], 50] (* Alonso del Arte, Nov 22 2011 *)

isok(m) = (m==1) || ((m%2) && ispower(m)); \\ Michel Marcus, May 04 2022

from sympy import mobius, integer_nthroot
def A075109(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 f(x): return int(n-1+x+sum(mobius(k)*((integer_nthroot(x,k)[0]+1>>1)-1) for k in range(2,x.bit_length())))
    return bisection(f,n,n) # Chai Wah Wu, Feb 25 2025

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} mu(k)*(1-zeta(k)*(2^k-1)/2^k) = 1.2890375574... - Amiram Eldar, Dec 19 2020

Definition clarified by N. J. A. Sloane, Dec 25 2021

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)

A353152 Perfect powers that are divisible by 5.

Original entry on oeis.org

25, 100, 125, 225, 400, 625, 900, 1000, 1225, 1600, 2025, 2500, 3025, 3125, 3375, 3600, 4225, 4900, 5625, 6400, 7225, 8000, 8100, 9025, 10000, 11025, 12100, 13225, 14400, 15625, 16900, 18225, 19600, 21025, 22500, 24025, 25600, 27000, 27225, 28900, 30625

Offset: 1

Marco Ripà, Apr 26 2022

a(n) == 0 (mod 25).

a(n) == {0, 25, 75} (mod 100).

			225 is a term since 225 = 15^2 is a power of a multiple of 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000351, A011557.
Intersection of A001597 and A008587.
Cf. A075090 (even perfect powers).

Select[Range[32000], And[GCD @@ FactorInteger[#][[All, 2]] > 1, Divisible[#, 5]] &] (* Michael De Vlieger, Apr 30 2022 *)

isok(k) = ispower(k) && !(k%5); \\ Michel Marcus, Apr 27 2022

upto(n) = { my(res = List()); forstep(i = 5, sqrtint(n), 5, c = i; for(e = 2, logint(n, i), c*=i; listput(res, c) ) ); Set(res) } \\ David A. Corneth, Apr 27 2022

a(n) = (5*m)^k for some positive m and some k > 1. - David A. Corneth, Apr 28 2022

Sum_{n>=1} 1/a(n) = -Sum_{k>=2} mu(k)*zeta(k)/5^k = 0.0756933503... - Amiram Eldar, Jul 02 2022

A123995 First occurrence of prime gaps which are perfect powers.

Original entry on oeis.org

2, 7, 89, 1831, 5591, 9551, 89689, 396733, 3851459, 11981443, 70396393, 202551667, 1872851947, 10958687879, 47203303159, 767644374817, 1999066711391, 8817792098461, 78610833115261, 497687231721157, 2069461000669981

Offset: 1

Walter Kehowski, Oct 31 2006

nonn

So far the powers have occurred in numerical order. Here is the list of primes and powers: [7, 4], [89, 8], [1831, 16], [5591, 32], [9551, 36], [89689, 64], [396733, 100], [3851459, 128], [11981443, 144], [70396393, 196], [202551667, 216], [1872851947, 256], [10958687879, 324]. I have searched out to the prime p=26689111613.
The old definition was confusing. What is meant was: primes p such that nextprime(p)-p is an element of A001597 (or A075090: even perfect powers, for n > 1), and p is the smallest prime followed by this gap. - M. F. Hasler, Oct 18 2018
A138198 is a subsequence. - M. F. Hasler, Oct 18 2018

			a(2)=89 since nextprime(89)-89=97-89=8 is the first occurrence of 8 as a difference between successive primes.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].

Cf. A080370, A113472, A000230, A001597 (perfect powers), A075090, A002386, A138198.

with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi end: P:={}; Q:=[]; p:=2; for w to 1 do for k from 0 do # keep track if k mod 10^6 = 0 then print(k,p) fi; lastprime:=p; q:=nextprime(p); d:=q-p; x:=egcd(d); if x>1 and not d in P then P:=P union {d}; Q:=[op(Q), [p,d]]; print(p,d); print(P); print(Q); fi ; p:=q; od od; # let it run with AutoSave enabled.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; perfectPowerQ[x_] := GCD @@ Last /@ FactorInteger@x > 1; dd = {1}; pp = {2}; qq = {3}; p = 3; Do[q = NextPrim@p; d = q - p; If[perfectPowerQ@d && ! MemberQ[dd, d], Print@q; AppendTo[pp, p]; AppendTo[dd, d]]; p = q, {n, 10^7}]; pp (* Robert G. Wilson v, Nov 03 2006 *)

S=[];print1(p=2);forprime(q=1+p,,ispower(q-p)&& !setsearch(S,q-p)&& !print1(","p)&& S=setunion(S,[q-p]);p=q) \\ M. F. Hasler, Oct 18 2018

Previous prime before A123996.

Edited and extended by Robert G. Wilson v, Nov 03 2006 and corrected Nov 04 2006

Better definition from M. F. Hasler, Oct 18 2018

A123996 Smallest prime q such that the gap between q and the previous prime is a perfect power that has not occurred earlier as a gap.

Original entry on oeis.org

3, 11, 97, 1847, 5623, 9587, 89753, 396833, 3851587, 11981587, 70396589, 202551883, 1872852203, 10958688203, 47203303559, 767644375301, 8817792099037, 78610833115937, 497687231721941, 2069461000670881

Offset: 1

Walter Kehowski, Oct 31 2006

nonn

So far the powers have occurred in numerical order. Here is the list of primes and powers: [11, 4], [97, 8], [1847, 16], [5623, 32], [9587, 36], [89753, 64], [396833, 100], [3851587, 128], [11981587, 144], [70396589, 196], [202551883, 216], [1872852203, 256], [10958688203, 324]. I have searched out to the prime p=26689111613.

			a(2)=97 since 97-prevprime(97)=97-89=8 is the first occurrence of 8 as a difference between successive primes.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].

Cf. A080370, A113472, A000230, A001597, A075090.

with(numtheory); egcd := proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi end: P:={}; Q:=[]; p:=2; for w to 1 do for k from 0 do # keep track if k mod 10^6 = 0 then print(k,p) fi; lastprime:=p; q:=nextprime(p); d:=q-p; x:=egcd(d); if x>1 and not d in P then P:=P union {d}; Q:=[op(Q), [q,d]]; print(q,d); print(P); print(Q); fi ; p:=q; od od; # let it run with AutoSave enabled.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; perfectPowerQ[x_] := GCD @@ Last /@ FactorInteger@x > 1; dd = {1}; pp = {2}; qq = {3}; p = 3; Do[q = NextPrim@p; d = q - p; If[perfectPowerQ@d && !MemberQ[dd, d], Print@q; AppendTo[qq, q]; AppendTo[dd, d]]; p = q, {n, 10^7}]; qq (* Robert G. Wilson v, Nov 03 2006 *)

Next prime after A123995.

Edited and extended by Robert G. Wilson v, Nov 03 2006, corrected Nov 04 2006

Definition corrected by M. F. Hasler, Oct 19 2018

cp's OEIS Frontend

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

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A075109 Odd perfect powers (1 together with numbers m^k, m odd, k >= 2).

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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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Mathematica

PARI

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A353152 Perfect powers that are divisible by 5.

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Mathematica

PARI

PARI

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A123995 First occurrence of prime gaps which are perfect powers.

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Maple

Mathematica

PARI

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A123996 Smallest prime q such that the gap between q and the previous prime is a perfect power that has not occurred earlier as a gap.

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