A025475 -id:A025475 - cp's OEIS frontend

A228018 Prime powers p (A025475) such that the sum of the proper divisors of p is also a prime power.

9, 49, 243, 961, 16129, 67092481, 17179607041, 274876858369, 4611686014132420609

Offset: 1

Alex Ratushnyak, Aug 02 2013

Numbers k such that both k and A001065(k) are in A025475.
Eight of the first nine terms are squares of Mersenne primes (A133049).
From Pontus von Brömssen, Sep 17 2024: (Start)
All squares of Mersenne primes are terms.
10^20 < a(10) <= A133049(9) = (2^61-1)^2.
All terms are odd. Otherwise, 2^k would be a term for some positive k >= 2 and then A001065(2^k) = 2^k-1 would be a prime power in A025475, which is impossible by Catalan's conjecture (Mihăilescu's theorem).
(End)

			Proper divisors of 243 are 1, 3, 9, 27, 81, their sum is 121 = 11^2, so 243 is in the sequence.

Cf. A025475, A001065, A133049.

#include 
#include 
#define TOP (1ULL<<32)
typedef unsigned long long U64;
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1== *(U64*)p2) return 0;
  return (*(U64*)p1 < *(U64*)p2) ? -1 : 1;
}
U64 findElement(U64 *a, U64 start, U64 end, U64 element) {
  if (start+1==end)  return (a[start]==element);
  U64 mid = (start+end)/2;
  if (a[mid] > element)
    return findElement(a, start, mid, element);
  return findElement(a, mid, end, element);
}
int main() {
  U64 i, j, p, n=0, *pp = (U64*)malloc(TOP/2), sum;
  unsigned char *c = (unsigned char *)malloc(TOP/16);
  if (!c || !pp) exit(1);
  memset(c, 0, TOP/16);
  pp[n++] = 1;
  for (i=1; i < TOP; i+=2) {
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      for (p=i+(i==1), j = p*p; ; j*=p) {
        pp[n++] = j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<60)*16.0)) break;
      }
      if (i>1)
        for (j=i*i>>1; j>3] |= 1<<(j&7);
    }
    if ((i&(i-2))==1)  printf("%llu ", i);
  }
  printf("// %llu\n\n", n);
  qsort(pp, n, 8, compare64);
  for (i=1; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0)
      for (p=i+(i==1), sum=1+p, j = p*p; ; j*=p) {
        if (findElement(pp, 0, n, sum)) printf("%llu, ", j);
        sum += j;
        double k = ((double)j) * ((double)p);
        if (k >= ((double)(1ULL<<60)*16.0)) break;
      }
  return 0;
}

for(n=1,10^6,if(!isprime(n),v=factor(n);if(matsize(v)[1]==1,s=sumdiv(n,d,d)-n;if(!isprime(s),vv=factor(s);if(matsize(vv)[1]==1,print(n)))))) /* Ralf Stephan, Aug 05 2013 */

A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

N. J. A. Sloane

The term "prime power" is ambiguous. To a mathematician it means any number p^k, p prime, k >= 0, including p^0 = 1.
Any nonzero integer is a product of primes and units, where the units are +1 and -1. This is tied to the Fundamental Theorem of Arithmetic which proves that the factorizations are unique up to order and units. (So, since 1 = p^0 does not have a well defined prime base p, it is sometimes not regarded as a prime power. See A246655 for the sequence without 1.)
These numbers are (apart from 1) the numbers of elements in finite fields. - Franz Vrabec, Aug 11 2004
Numbers whose divisors form a geometrical progression. The divisors of p^k are 1, p, p^2, p^3, ..., p^k. - Amarnath Murthy, Jan 09 2002
These are also precisely the orders of those finite affine planes that are known to exist as of today. (The order of a finite affine plane is the number of points in an arbitrarily chosen line of that plane. This number is unique for all lines comprise the same number of points.) - Peter C. Heinig (algorithms(AT)gmx.de), Aug 09 2006
Except for first term, the index of the second number divisible by n in A002378, if the index equals n. - Mats Granvik, Nov 18 2007
These are precisely the numbers such that lcm(1,...,m-1) < lcm(1,...,m) (=A003418(m) for m>0; here for m=1, the l.h.s. is taken to be 0). We have a(n+1)=a(n)+1 if a(n) is a Mersenne prime or a(n)+1 is a Fermat prime; the converse is true except for n=7 (from Catalan's conjecture) and n=1, since 2^1-1 and 2^0+1 are not considered as Mersenne resp. Fermat prime. - M. F. Hasler, Jan 18 2007, Apr 18 2010
The sequence is A000015 without repetitions, or more formally, A000961=Union[A000015]. - Zak Seidov, Feb 06 2008
Except for a(1)=1, indices for which the cyclotomic polynomial Phi[k] yields a prime at x=1, cf. A020500. - M. F. Hasler, Apr 04 2008
Also, {A138929(k) ; k>1} = {2*A000961(k) ; k>1} = {4,6,8,10,14,16,18,22,26,32,34,38,46,50,54,58,62,64,74,82,86,94,98,...} are exactly the indices for which Phi[k](-1) is prime. - M. F. Hasler, Apr 04 2008
A143201(a(n)) = 1. - Reinhard Zumkeller, Aug 12 2008
Number of distinct primes dividing n=omega(n) < 2. - Juri-Stepan Gerasimov, Oct 30 2009
Numbers n such that Sum_{p-1|p is prime and divisor of n} = Product_{p-1|p is prime and divisor of n}. A055631(n) = A173557(n-1). - Juri-Stepan Gerasimov, Dec 09 2009, Mar 10 2010
Numbers n such that A028236(n) = 1. Klaus Brockhaus, Nov 06 2010
A188666(k) = a(k+1) for k: 2*a(k) <= k < 2*a(k+1), k > 0; notably a(n+1) = A188666(2*a(n)). - Reinhard Zumkeller, Apr 25 2011
A003415(a(n)) = A192015(n); A068346(a(n)) = A192016(n); a(n)=A192134(n) + A192015(n). - Reinhard Zumkeller, Jun 26 2011
A089233(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2013
The positive integers n such that every element of the symmetric group S_n which has order n is an n-cycle. - W. Edwin Clark, Aug 05 2014
Conjecture: these are numbers m such that Sum_{k=0..m-1} k^phi(m) == phi(m) (mod m), where phi(m) = A000010(m). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
Numbers whose (increasingly ordered) divisors are alternatingly squares and nonsquares. - Michel Marcus, Jan 16 2019
Possible numbers of elements in a finite vector space. - Jianing Song, Apr 22 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
M. Koecher and A. Krieg, Ebene Geometrie, Springer, 1993.
R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge 1986, Theorem 2.5, p. 45.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Brady Haran and Günter Ziegler, Cannons and Sparrows, Numberphile video (2018).
Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Projective Plane
Index entries for "core" sequences

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). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018
Cf. A008480, A010055, A065515, A095874, A025473.
Cf. indices of record values of A003418; A000668 and A019434 give a member of twin pairs a(n+1)=a(n)+1.
A138929(n) = 2*a(n).
Cf. A000040, A001221, A001477. - Juri-Stepan Gerasimov, Dec 09 2009
A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j). - Klaus Brockhaus, Nov 06 2010
A000015(n) = Min{term : >= n}; A031218(n) = Max{term : <= n}.
Complementary (in the positive integers) to sequence A024619. - Jason Kimberley, Nov 10 2015

import Data.Set (singleton, deleteFindMin, insert)
a000961 n = a000961_list !! (n-1)
a000961_list = 1 : g (singleton 2) (tail a000040_list) where
g s (p:ps) = m : g (insert (m * a020639 m) $ insert p s') ps
where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012, Apr 25 2011

[1] cat [ n : n in [2..250] | IsPrimePower(n) ]; // corrected by Arkadiusz Wesolowski, Jul 20 2012

readlib(ifactors): for n from 1 to 250 do if nops(ifactors(n)[2])=1 then printf(`%d,`,n) fi: od:
# second Maple program:
a:= proc(n) option remember; local k; for k from
      1+a(n-1) while nops(ifactors(k)[2])>1 do od; k
    end: a(1):=1: A000961:= a:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2013

Select[ Range[ 2, 250 ], Mod[ #, # - EulerPhi[ # ] ] == 0 & ]
Select[ Range[ 2, 250 ], Length[FactorInteger[ # ] ] == 1 & ]
max = 0; a = {}; Do[m = FactorInteger[n]; w = Sum[m[[k]][[1]]^m[[k]][[2]], {k, 1, Length[m]}]; If[w > max, AppendTo[a, n]; max = w], {n, 1, 1000}]; a (* Artur Jasinski *)
Join[{1}, Select[Range[2, 250], PrimePowerQ]] (* Jean-François Alcover, Jul 07 2015 *)

A000961(n,l=-1,k=0)=until(n--<1,until(lA000961(lim=999,l=-1)=for(k=1,lim, l==lcm(l,k) && next; l=lcm(l,k); print1(k,",")) \\ M. F. Hasler, Jan 18 2007

isA000961(n) = (omega(n) == 1 || n == 1) \\ Michael B. Porter, Sep 23 2009

nextA000961(n)=my(m,r,p);m=2*n;for(e=1,ceil(log(n+0.01)/log(2)),r=(n+0.01)^(1/e);p=prime(primepi(r)+1);m=min(m,p^e));m \\ Michael B. Porter, Nov 02 2009

is(n)=isprimepower(n) || n==1 \\ Charles R Greathouse IV, Nov 20 2012

list(lim)=my(v=primes(primepi(lim)),u=List([1])); forprime(p=2,sqrtint(lim\1),for(e=2,log(lim+.5)\log(p),listput(u,p^e))); vecsort(concat(v,Vec(u))) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import primerange
def A000961_list(limit): # following Python style, list terms < limit
    L = [1]
    for p in primerange(1, limit):
        pe = p
        while pe < limit:
            L.append(pe)
            pe *= p
    return sorted(L) # Chai Wah Wu, Sep 08 2014, edited by M. F. Hasler, Jun 16 2022

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A000961(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

def A000961_list(n):
    R = [1]
    for i in (2..n):
        if i.is_prime_power(): R.append(i)
    return R
A000961_list(227) # Peter Luschny, Feb 07 2012

a(n) = A025473(n)^A025474(n). - David Wasserman, Feb 16 2006
a(n) = A117331(A117333(n)). - Reinhard Zumkeller, Mar 08 2006
Panaitopol (2001) gives many properties, inequalities and asymptotics, including a(n) ~ prime(n). - N. J. A. Sloane, Oct 31 2014, corrected by M. F. Hasler, Jun 12 2023 [The reference gives pi*(x) = pi(x) + pi(sqrt(x)) + ... where pi*(x) counts the terms up to x, so it is the inverse function to a(n).]
m=a(n) for some n <=> lcm(1,...,m-1) < lcm(1,...,m), where lcm(1...0):=0 as to include a(1)=1. a(n+1)=a(n)+1 <=> a(n+1)=A019434(k) or a(n)=A000668(k) for some k (by Catalan's conjecture), except for n=1 and n=7. - M. F. Hasler, Jan 18 2007, Apr 18 2010
A001221(a(n)) < 2. - Juri-Stepan Gerasimov, Oct 30 2009
A008480(a(n)) = 1 for all n >= 1. - Alois P. Heinz, May 26 2018
Sum_{k=1..n} 1/a(k) ~ log(log(a(n))) + 1 + A077761 + A136141. - François Huppé, Jul 31 2024

Description modified by Ralf Stephan, Aug 29 2014

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

A246655 Prime powers: numbers of the form p^k where p is a prime and k >= 1.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211

Offset: 1

Peter Luschny and Franklin T. Adams-Watters, Sep 01 2014

The elements are called prime powers in contrast to the powers of primes which are the numbers of the same form but with k >= 0, cf. A000961.
Every nonzero integer is the product of elements of this sequence which are relatively prime and an element of {-1, 1}. This product is up to a rearrangement of the factors unique. (This statement is the fundamental theorem of arithmetic.)
These numbers are the numbers such that the von Mangoldt function is nonzero.
These numbers are the numbers of elements in finite fields. - Franz Vrabec, Aug 11 2004
A positive integer n is a prime power if and only if nZ is a primary ideal of Z. - John Cremona, Sep 02 2014
Also, numbers n divisible by their cototients A051953(n). - Ivan Neretin, May 29 2016
Numbers n such that (theta_3(q) - theta_3(q^n)) / 2 is the g.f. of a multiplicative sequence. - Michael Somos, Oct 17 2016
Numbers that are evenly divisible by exactly one prime number. - Lee A. Newberg, May 07 2018
Ram proved that these are precisely the numbers n such that the binomial coefficients n!/(m!(n-m)!) for m = 1..n-1 have a common factor greater than 1 (which is the unique prime dividing n). See Joris, Oestreicher & Steinig for a generalization. - Charles R Greathouse IV, Apr 24 2019
Blagojević & Ziegler prove that for these n and for any convex polygon in the plane, the polygon can be partitioned into n polygons with equal area and equal perimeter. The result is conjectured (by Nandakumar & Rao, who proved the case n = 2) to hold for all n. - Charles R Greathouse IV, Apr 24 2019
Numbers n such that A367064(n) < 0. - Chai Wah Wu, Nov 06 2023
This sequence represents all positive high amplitude peaks of the inverse Riemann spectrum R(x)= Sum_{k=1..oo} -cos(log(x)*Im(z_k)) going over the imaginary part of the nontrivial zeros "Im(z_k)" in the Riemann zeta function. - Marc Morgenegg, Jul 29 2025

Robert Israel, Table of n, a(n) for n = 1..10000
Brady Haran and Günter Ziegler, Cannons and Sparrows, Numberphile video (2018)
H. Joris, C. Oestreicher and J. Steinig, The greatest common divisor of certain sets of binomial coefficients, Journal of Number Theory 21 (1985), pp. 101-119.
Pavle V. M. Blagojević and Günter M. Ziegler, Convex equipartitions via equivariant obstruction theory, arXiv:1202.5504 [math.AT], 2012-2013; Israel Journal of Mathematics 200:1 (June 2014), pp 49-77.
Karl-Heinz Kuhl, Inverse Riemann Spectrum
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.
Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
Balak Ram, Common factors of n!/(m!(n-m)!), (m = 1, 2, ... n-1), Journal of the Indian Mathematical Club (Madras) 1 (1909), pp. 39-43.
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Projective Plane
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.
Wikipedia, Prime power
Index entries for "core" sequences

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). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018
Cf. A001221, A014963, A069513, A367064.
Partial sums of A275120.

select(t -> nops(numtheory:-factorset(t))=1, [$1..1000]); # Robert Israel, Sep 01 2014
A246655 := proc(n) A000961(n+1) end proc: # R. J. Mathar, Jan 09 2017
isprimepower := n -> nops(NumberTheory:-PrimeFactors(n)) = 1: # Peter Luschny, Oct 09 2022

Mathematica
```
Select[Range[222], PrimePowerQ]
```
PARI
```
[p| p <- [1..222], isprimepower(p)]
```

list(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023

from sympy import primerange
m = 10**5
A246655 = []
for p in primerange(1,m):
    pe = p
    while pe < m:
        A246655.append(pe)
        pe *= p
A246655 = sorted(A246655) # Chai Wah Wu, Sep 04 2014

from sympy import primepi, integer_nthroot
def A246655(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,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 20 2024

[n for n in (1..222) if sloane.A001221(n) == 1]

a(n) is characterized by A001221(a(n)) = 1.
a(n) is characterized by A014963(a(n)) != 1.
Euler's A000010(a(n)) = a(n)*(1 - 1/A014963(a(n))).
All three relations above are not true for A000961(n) instead of a(n).
Sum_{k=1..n} 1/a(k) ~ log(log(a(n))) + A077761 + A136141. - François Huppé, Jul 31 2024

A057820 First differences of sequence of consecutive prime powers (A000961).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, 4, 2, 6, 2, 2, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 6, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 8, 5, 1, 6, 6, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 6, 6, 4, 2, 4, 6, 2, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Labos Elemer, Nov 08 2000

nonn

a(n) = 1 iff A000961(n) = A006549(k) for some k. - Reinhard Zumkeller, Aug 25 2002
Also run lengths of distinct terms in A070198. - Reinhard Zumkeller, Mar 01 2012
Does this sequence contain all positive integers? - Gus Wiseman, Oct 09 2024

			Odd differences arise in pairs in neighborhoods of powers of 2, like {..,2039,2048,2053,..} gives {..,11,5,..}

Michael B. Porter, Table of n, a(n) for n = 1..10000

Cf. A000961, A036616, A001223.
For perfect-powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
Positions of ones are A375734.
Run-compression is A376308.
Run-lengths are A376309.
Sorted positions of first appearances are A376340.
The second (instead of first) differences are A376596, zeros A376597.
Prime-powers:
- terms: A000961 or A246655, complement A024619
- differences: A057820 (this), first appearances A376341
- runs: A373675, A373673, A373674, A174965
- anti-runs: A373576, A120430, A006549, A373671
Non-prime-powers:
- terms: A361102
- differences: A375708 (ones A375713)
- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670)
- anti-runs: A373679, A373575, A255346, A373672
Cf. A000015, A014210, A014963, A025475, A025528, A027833, A037201, A046933, A076259, A078147, A093555, A345531, A376310.

a057820_list = zipWith (-) (tail a000961_list) a000961_list
-- Reinhard Zumkeller, Mar 01 2012

A057820 := proc(n)
        A000961(n+1)-A000961(n) ;
end proc: # R. J. Mathar, Sep 23 2016

Map[Length, Split[Table[Apply[LCM, Range[n]], {n, 1, 150}]]] (* Geoffrey Critzer, May 29 2015 *)
Join[{1},Differences[Select[Range[500],PrimePowerQ]]] (* Harvey P. Dale, Apr 21 2022 *)

isA000961(n) = (omega(n) == 1 || n == 1)
n_prev=1;for(n=2,500,if(isA000961(n),print(n-n_prev);n_prev=n)) \\ Michael B. Porter, Oct 30 2009

from sympy import primepi, integer_nthroot
def A057820(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    r, k = m, f(m)+1
    while r != k: r, k = k, f(k)+1
    return r-m # Chai Wah Wu, Sep 12 2024

a(n) = A000961(n+1) - A000961(n).

Offset corrected and b-file adjusted by Reinhard Zumkeller, Mar 03 2012

A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 2, 3, 3, 3, 3, 1, 3, 3, 4, 1, 3, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 3, 1, 4, 3, 3, 1, 4, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 4, 1, 4, 4, 3, 1, 3, 1, 4, 3

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

The fixed points of the x -> A181819(x) map are 1 and 2. Note that the x -> A000005(x) map has the same fixed points, and that A000005(n) = A181819(n) iff n is cubefree (cf. A004709). Under the x -> A181819(x) map, it seems significantly easier to generalize about which kinds of integers take a given number of iterations to reach a fixed point than under the x -> A000005(x) map.

Also the number of steps in the reduction of the multiset of prime factors of n wherein one repeatedly takes the multiset of multiplicities. For example, the a(90) = 5 steps are {2,3,3,5} -> {1,1,2} -> {1,2} -> {1,1} -> {2} -> {1}. - Gus Wiseman, May 13 2018

			A181819(6) = 4; A181819(4) = 3; A181819(3) = 2; A181819(2) = 2. Therefore, a(6) = 3, a(4) = 2, a(3)= 1, and a(2) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

A182857 gives values of n where a(n) increases to a record.

Cf. A000961, A001222, A003434, A005117, A007916, A036459, A112798, A130091, A181819, A182851-A182858, A238748, A304455, A304464, A304465.

a182850 n = length $ takeWhile (`notElem` [1,2]) $ iterate a181819 n
-- Reinhard Zumkeller, Mar 26 2012

Table[If[n<=2,0,Length[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]-1],{n,100}] (* Gus Wiseman, May 13 2018 *)

;; With memoization-macro definec.
(definec (A182850 n) (if (<= n 2) 0 (+ 1 (A182850 (A181819 n))))) ;; Antti Karttunen, Feb 05 2016

For n > 2, a(n) = a(A181819(n)) + 1.
a(n) = 0 iff n equals 1 or 2.
a(n) = 1 iff n is an odd prime (A000040(n) for n>1).
a(n) = 2 iff n is a composite perfect prime power (A025475(n) for n>1).
a(n) = 3 iff n is a squarefree composite integer or a power of a squarefree composite integer (cf. A182853).
a(n) = 4 iff n's prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number (cf. A182854).

A246547 Prime powers p^e where p is a prime and e >= 2 (prime powers without the primes or 1).

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 8192, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641

Offset: 1

Joerg Arndt, Aug 29 2014

These are sometimes called the proper prime powers.
A proper subset of A001597.
Equals A000961 \ A008578 = { x in A001597 | A001221(x)=1 }. - M. F. Hasler, Aug 29 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Essentially the same as A025475.
Cf. A000961, A001597, A025528, A051953, A136141, A246655.
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). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018

isA246547 := proc(n)
    local ifs;
    ifs := ifactors(n)[2] ;
    if nops(ifs) <> 1 then
        false;
    else
        is(op(2, op(1, ifs)) > 1);
    end if;
end proc:
for n from 2 do
    if isA246547(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Feb 01 2016 # Or:
isA246547 := n -> not isprime(n) and nops(numtheory:-factorset(n)) = 1:
select(isA246547, [$1..10000]); # Peter Luschny, May 01 2018

With[{upto=15000},Complement[Select[Range[upto],PrimePowerQ],Prime[ Range[ PrimePi[ upto]]]]] (* Harvey P. Dale, Nov 28 2014 *)
Select[ Range@ 15000, PrimePowerQ@# && !SquareFreeQ@# &] (* Robert G. Wilson v, Dec 01 2014 *)
With[{n = 15000}, Union@ Flatten@ Table[Array[p^# &, Floor@ Log[p, n] - 1, 2], {p, Prime@ Range@ PrimePi@ Sqrt@ n}] ] (* Michael De Vlieger, Jul 06 2018, faster program *)

for(n=1,10^5,if(isprimepower(n)>=2,print1(n,", ")));

m=10^5; v=[]; forprime(p=2, sqrtint(m), e=2; while(p^e<=m, v=concat(v, p^e); e++)); v=vecsort(v) \\ Faster program. Jens Kruse Andersen, Aug 29 2014

from sympy import primepi, integer_nthroot
def A246547(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x,k)[0]) 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 14 2024

def A246547List(n):
    return [p for p in srange(2, n) if p.is_prime_power() and not p.is_prime()]
print(A246547List(14642))  # Peter Luschny, Sep 16 2023

a(n) = A025475(n+1). - M. F. Hasler, Aug 29 2014

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Dec 21 2020

A053289 First differences of consecutive perfect powers (A001597).

Original entry on oeis.org

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

A080101 Number of prime powers in all composite numbers between n-th prime and next prime.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 28 2003

nonn

The maximum value of terms in the sequence, through the (10^5)th term, is 2. - Harvey P. Dale, Aug 24 2014

This is conjectured to be the maximum, see also A366833. - Gus Wiseman, Nov 06 2024

			There are two prime powers between 2179 = A000040(327) and 2203 = A000040(328): 2187 = 3^7 and 2197 = 13^3, therefore a(327) = 2, A080102(327) = 2187 and A080103(327) = 2197.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A080102, A080103, A025475, A000961.
For powers of 2 instead of primes we have A244508, see also A013597, A014210, A014234, A304521.
Adding one gives A366833.
For non-prime-powers instead of prime-powers we have A368748.
Positions of positive terms are A377057, primes A053607.
Positions of 0 are A377286.
Positions of 1 are A377287.
Positions of 2 are A377288, primes A053706.
For perfect-powers (instead of prime-powers) we have A377432.
A000015 gives the least prime-power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820, seconds A376596.
A031218 gives the greatest prime-power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A065514 gives the greatest prime-power < prime(n), difference A377289.
A246655 lists the prime-powers not including 1, complement A361102.
A345531 gives the least prime-power > prime(n), difference A377281.
Cf. A001597, A002808, A024619, A065890, A182908, A224363, A376597, A377051, A377054, A377436.

a := proc(n) local c, k, p: c, p := 0, ithprime(n): for k from p+1 to nextprime(p)-1 do if nops(numtheory:-factorset(k)) = 1 then c := c+1: fi: od: c: end:
seq(a(n), n = 1 .. 105); # Lorenzo Sauras Altuzarra, Jul 08 2022

prpwQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]==1&&fi[[1,2]]>1]; nn=600;With[{pwrs=Table[If[prpwQ[n],1,0],{n,nn}]},Table[Total[ Take[ pwrs,{Prime[n],Prime[n+1]}]],{n,PrimePi[nn]-1}]] (* Harvey P. Dale, Aug 24 2014 *)
Table[Length[Select[Range[Prime[n]+1,Prime[n+1]-1],PrimePowerQ]],{n,30}] (* Gus Wiseman, Nov 06 2024 *)

a(n) = A366833(n) - 1. - Gus Wiseman, Nov 06 2024

cp's OEIS Frontend

A226035 Prime powers (A025475) whose decimal digit reversal is also a prime power.

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A228018 Prime powers p (A025475) such that the sum of the proper divisors of p is also a prime power.

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A000961 Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).

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

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A246655 Prime powers: numbers of the form p^k where p is a prime and k >= 1.

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A057820 First differences of sequence of consecutive prime powers (A000961).

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