A001597 -id:A001597 - cp's OEIS frontend

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

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

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

A000533 a(0)=1; a(n) = 10^n + 1, n >= 1.

Original entry on oeis.org

1, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001, 1000000000001, 10000000000001, 100000000000001, 1000000000000001, 10000000000000001

Offset: 0

N. J. A. Sloane

Also, b^n+1 written in base b, for any base b >= 2.
Also, A083318 written in base 2. - Omar E. Pol, Feb 24 2008, Dec 30 2008
Also, palindromes formed from the reflected decimal expansion of the concatenation of 1 and infinite 0's. - Omar E. Pol, Dec 14 2008
It seems that the sequence gives 'all' positive integers m such that m^4 is a palindrome. Note that a(0)^4 = 1 is a palindrome and for n > 0, a(n)^4 = (10^n + 1)^4 = 10^(4n) + 4*10^(3n) + 6*10^(2n) + 4*10^(n) + 1 is a palindrome. - Farideh Firoozbakht, Oct 28 2014
a(n)^2 starts with a(n)+1 for n >= 1. - Dhilan Lahoti, Aug 31 2015
From Peter Bala, Sep 25 2015: (Start)
The simple continued fraction expansion of sqrt(a(2*n)) = [10^n; 2*10^n, 2*10^n, ...] has period 1.
The simple continued fraction expansion of sqrt(a(6*n))/a(2*n) = [10^n - 1; 1, 10^n - 1, 10^n - 1, 1, 2*(10^n - 1), ...] has period 5.
The simple continued fraction expansion of sqrt(a(10*n))/a(2*n) = [10^(3*n) - 10^n; 10^n, 10^n, 2*(10^(3*n) - 10^n), ...] has period 3.
As n increases, these expansions have large partial quotients.
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions.
Empirically, we also see exceptionally large partial quotients in the continued fraction expansions of the m-th root of the numbers a(m*n), for m >= 3. For example, it appears that the continued fraction expansion of a(3*n)^(1/3) begins [10^n; 3*10^(2*n), 10^n, 4.5*10^(2*n), 0.8*10^n, ( 9*10^(2*n + 2) - 144 + 24*(2^mod(n,3) - 1) )/168, ...]. As n increases, the expansion begins with 6 large partial quotients. An example is given below. Cf. A002283, A066138 and A168624.
(End)
a(1) and a(2) are the only prime terms up to n=100000. - Daniel Arribas, Jun 04 2016
Based on factors from A001271, the first abundant number in this sequence should occur in the first M terms, where M is the double factorial M=7607!!. Is any abundant number known in this sequence? - Sergio Pimentel, Oct 04 2019
The (3^5 * 5^2 * 7^2 * 11^2 * 13^2 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47 * 53 * 59 * 61 * 67 * 71 * 73 * 79 * 83 * 89 * 97 * 101 * 103 * 107 * 109 * 113 * 127 * 131 * 137 * 139 * 157 * 163 * 181 * 191 * 241 * 251 * 263)-th term of this sequence is an abundant number. - Jon E. Schoenfield, Nov 19 2019
The only perfect power (i.e., a perfect square, a perfect cube, and so forth) in the present sequence is a(0) by Mihăilescu's theorem (since 10^n is a perfect power not equal to 2^3 - see the links in A001597). - Marco Ripà, Feb 04 2025

			The continued fraction expansion of a(9)^(1/3) begins [1000; 3000000, 1000, 4500000, 800, 5357142, 1, 6, 14, 6, 1, 5999999, 6, 1, 12, 7, 1, ...] with 5 large partial quotients immediately following the integer part of the number. - _Peter Bala_, Sep 25 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..100
John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 2019.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002283, A066138, A083318, A138144, A138145, A152756, A168624.

[10^n + 1 - 0^n: n in [0..30]]; // Vincenzo Librandi, Jul 15 2011

Join[{1},LinearRecurrence[{11,-10},{11,101},20]] (* Harvey P. Dale, May 01 2014 *)

a(n)=if(n,10^n+1,1) \\ Charles R Greathouse IV, Oct 28 2014

a(n) = 10^n + 1 - 0^n. - Reinhard Zumkeller, Jun 10 2003
From Paul Barry, Feb 05 2005: (Start)
G.f.: (1-10*x^2)/((1-x)*(1-10*x));
a(n) = Sum_{k=0..n} binomial(n, k)*0^(k(n-k))*10^k. (End)
a(n) = A178500(n) + 1. - Reinhard Zumkeller, May 28 2010
E.g.f.: exp(x) + exp(10*x) - 1. - Ilya Gutkovskiy, Jun 03 2016

A007916 Numbers that are not perfect powers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

R. Muller

From Gus Wiseman, Oct 23 2016: (Start)
There is a 1-to-1 correspondence between integers N >= 2 and sequences a(x_1),a(x_2),...,a(x_k) of terms from this sequence. Every N >= 2 can be written uniquely as a "power tower"
N = a(x_1)^a(x_2)^a(x_3)^...^a(x_k),
where the exponents are to be nested from the right.
Proof: If N is not a perfect power then N = a(x) for some x, and we are done. Otherwise, write N = a(x_1)^M for some M >=2, and repeat the process. QED
Of course, prime numbers also have distinct power towers (see A164336). (End)
These numbers can be computed with a modified Sieve of Eratosthenes: (1) start at n=2; (2) if n is not crossed out, then append n to the sequence and cross out all powers of n; (3) set n = n+1 and go to step 2. - Sam Alexander, Dec 15 2003
These are all numbers such that the multiplicities of the prime factors have no common divisor. The first number in the sequence whose prime multiplicities are not coprime is 180 = 2 * 2 * 3 * 3 * 5. Mathematica: CoprimeQ[2,2,1]->False. - Gus Wiseman, Jan 14 2017

			Example of the power tower factorizations for the first nine positive integers: 1=1, 2=a(1), 3=a(2), 4=a(1)^a(1), 5=a(3), 6=a(4), 7=a(5), 8=a(1)^a(2), 9=a(2)^a(1). - _Gus Wiseman_, Oct 20 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..9875
Joakim Munkhammar, The Riemann zeta function as a sum of geometric series, The Mathematical Gazette (2020) Vol. 104, Issue 561, 527-530.
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993
Index entries for sequences generated by sieves

Complement of A001597. Union of A052485 and A052486.
Cf. A144338, A277562, A277564, A075802.
Cf. A153158 (squares of these numbers).
See A277562, A277564, A277576, A277615 for more about the power towers.
A278029 is a left inverse.
Cf. A052409.

a007916 n = a007916_list !! (n-1)
a007916_list = filter ((== 1) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

[n : n in [2..1000] | not IsPower(n) ];

Maple
```
See link.
```

a = {}; Do[If[Apply[GCD, Transpose[FactorInteger[n]][[2]]] == 1, a = Append[a, n]], {n, 2, 200}];
Select[Range[2,200],GCD@@FactorInteger[#][[All,-1]]===1&] (* Michael De Vlieger, Oct 21 2016. Corrected by Gus Wiseman, Jan 14 2017 *)

is(n)=!ispower(n)&&n>1 \\ Charles R Greathouse IV, Jul 01 2013

from sympy import mobius, integer_nthroot
def A007916(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 13 2024

A075802(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2009
Gcd(exponents in prime factorization of a(n)) = 1, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
a(n) ~ n. - Charles R Greathouse IV, Jul 01 2013
A052409(a(n)) = 1. - Ridouane Oudra, Nov 23 2024

More terms from Henry Bottomley, Sep 12 2000
Edited by Charles R Greathouse IV, Mar 18 2010
Further edited by N. J. A. Sloane, Nov 09 2016

A025475 1 and the prime powers p^m where m >= 2, thus excluding the primes.

Original entry on oeis.org

1, 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

Offset: 1

David W. Wilson

Also nonprime n such that sigma(n)*phi(n) > (n-1)^2. - Benoit Cloitre, Apr 12 2002
If p is a term of the sequence, then the index n for which a(n) = p is given by n := b(p) := 1 + Sum_{k>=2} PrimePi(p^(1/k)). Here, the sum has floor(log_2(p)) positive terms. For any m > 0, the greatest number n such that a(n) <= m is also given by b(m), thus, b(m) is the number of such prime powers <= m. - Hieronymus Fischer, May 31 2013
That 8 and 9 are the only two consecutive integers in this sequence is known as Catalan's Conjecture and was proved in 2002 by Preda Mihăilescu. - Geoffrey Critzer, Nov 15 2015

T. D. Noe, Table of n, a(n) for n = 1..10000
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 4, 2013.
Preda Mihăilescu, On Catalan's Conjecture, Kuwait Foundation Lecture 30 - April 28, 2003.
Eric Weisstein's World of Mathematics, Prime Power.

Subsequence of A000961. - Reinhard Zumkeller, Jun 22 2011
Cf. A001597, A000720, A068982, A136141, A193166.
Differences give A053707.
Cf. A076048 (number of terms < 10^n).
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

a025475 n = a025475_list !! (n-1)
a025475_list = filter ((== 0) . a010051) a000961_list
-- Reinhard Zumkeller, Jun 22 2011

isA025475 := proc(n)
    if n < 1 then
        false;
    elif n = 1 then
        true;
    elif isprime(n) then
        false;
    elif nops(numtheory[factorset](n)) = 1 then
        true;
    else
        false;
    end if;
end proc:
A025475 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA025475(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
# R. J. Mathar, Jun 06 2013
# alternative:
N:= 10^5: # to get all terms <= N
Primes:= select(isprime, [2,(2*i+1 $ i = 1 .. floor((sqrt(N)-1)/2))]):
sort([1,seq(seq(p^i, i=2..floor(log[p](N))),p=Primes)]); # Robert Israel, Jul 27 2015

A025475 = Select[ Range[ 2, 10000 ], ! PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ] ] == 0 & ]
A025475 = Sort[ Flatten[ Table[ Prime[n]^i, {n, 1, PrimePi[ Sqrt[10^4]]}, {i, 2, Log[ Prime[n], 10^4]}]]]
{1}~Join~Select[Range[10^4], And[! PrimeQ@ #, PrimePowerQ@ #] &] (* Michael De Vlieger, Jul 04 2016 *)
Join[{1},Select[Range[100000],PrimePowerQ[#]&&!PrimeQ[#]&]] (* Harvey P. Dale, Oct 29 2023 *)

for(n=1,10000,if(sigma(n)*eulerphi(n)*(1-isprime(n))>(n-1)^2,print1(n,",")))

is_A025475(n)={ ispower(n,,&p) && isprime(p) || n==1 }  \\ M. F. Hasler, Sep 25 2011

list(lim)=my(v=List([1]),L=log(lim+.5));forprime(p=2,(lim+.5)^(1/3),for(e=3,L\log(p),listput(v,p^e))); vecsort(concat(Vec(v), apply(n->n^2,primes(primepi(sqrtint(lim\1)))))) \\ Charles R Greathouse IV, Nov 12 2012

list(lim)=my(v=List([1])); for(m=2,logint(lim\=1,2), forprime(p=2,sqrtnint(lim,m), listput(v, p^m))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

from sympy import primerange
A025475_list, m = [1], 10*2
m2 = m**2
for p in primerange(1,m):
    a = p**2
    while a < m2:
        A025475_list.append(a)
        a *= p
A025475_list = sorted(A025475_list) # Chai Wah Wu, Sep 08 2014

from sympy import primepi, integer_nthroot
def A025475(n):
    if n==1: return 1
    def f(x): return int(n-2+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 13 2024

The number of terms <= N is O(sqrt(N)*log N). [See Weisstein link] - N. J. A. Sloane, May 27 2022
A005171(a(n))*A010055(a(n)) = 1. - Reinhard Zumkeller, Nov 01 2009
A192280(a(n)) = 0 for n > 1. - Reinhard Zumkeller, Aug 26 2011
A014963(a(n)) - A089026(a(n)) = A014963(a(n)) - 1. - Eric Desbiaux, May 18 2013
From Hieronymus Fischer, May 31 2013: (Start)
The greatest number n such that a(n) <= m is given by 1 + Sum_{k>=2} A000720(floor(m^(1/k))).
Example 1:  m = 10^10 ==> n = 10085;
Example 2:  m = 10^11 ==> n = 28157;
Example 3:  m = 10^12 ==> n = 80071;
Example 4:  m = 10^15 ==> n = 1962690. (End)
Sum_{n>=2} 1/a(n) = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Oct 11 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=2} (1 + 1/a(n)) = Product_{k>=2} zeta(k)/zeta(2*k) = 2.0729553047...
Product_{n>=2} (1 - 1/a(n)) = A068982. (End)

Edited by Daniel Forgues, Aug 18 2009

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

A007489 a(n) = Sum_{k=1..n} k!.

Original entry on oeis.org

0, 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, 43954713, 522956313, 6749977113, 93928268313, 1401602636313, 22324392524313, 378011820620313, 6780385526348313, 128425485935180313, 2561327494111820313, 53652269665821260313, 1177652997443428940313

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Equals row sums of triangle A143122 starting (1, 3, 9, 33, ...). - Gary W. Adamson, Jul 26 2008
a(n) for n>=4 is never a perfect square. - Alexander R. Povolotsky, Oct 16 2008
Number of cycles that can be written in the form (j,j+1,j+2,...), in all permutations of {1,2,...,n}. Example: a(3)=9 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132) we have 3+2+2+1+1+0=9 such cycles. - Emeric Deutsch, Jul 14 2009
Conjectured to be the length of the shortest word over {1,...,n} that contains each of the n! permutations as a factor (cf. A180632) [see Johnston]. - N. J. A. Sloane, May 25 2013
The above conjecture has been disproven for n>=6. See A180632 and the Houston 2014 reference. - Dmitry Kamenetsky, Mar 07 2016
a(n) is also the number of compositions of n if cardinal values do not matter but ordinal rankings do. Since cardinal values do not matter, a sequence of k summands summing to n can be represented as (s(1),...,s(k)), where the s's are positive integers and the numbers in parentheses are the initial ordinal rankings. The number of compositions of these summands are equal to k!, with k ranging from 1 to n. - Gregory L. Simay, Jul 31 2016
When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the left. Compare array A211370 for circular shifts to the left in a broader sense. Compare sequence A001563 for circular shifts to the right. - Tilman Piesk, Apr 29 2017
Since a(n) = (1!+2!+3!+...+n!) = 3(1+3!/3+4!/3+...+n!/3) is a multiple of 3 for n>2, the only prime in this sequence is a(2) = 3. - Eric W. Weisstein, Jul 15 2017
Generalization of 2nd comment: a(n) for n>=4 is never a perfect power (A007916) (Chentzov link). - Bernard Schott, Jan 26 2023

			a(4) = 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. - _Michael B. Porter_, Aug 03 2016

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Section B44, Springer 2010.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Carauleanu Marc, Table of n, a(n) for n = 0..212 (first 100 terms from T. D. Noe)
N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics, problem 115, pp. 28 and 201-202, Dover publications, Inc., New York, 1993.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Robin Houston, Tackling the Minimal Superpermutation Problem, arXiv:1408.5108 [math.CO], 2014.
Nathaniel Johnston, The minimal superpermutation problem (2013)
Nathaniel Johnston, Non-uniqueness of minimal superpermutations, Discrete Math. 313 (2013), no. 14, 1553--1557. MR3047396
S. Legendre and P. Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Eric Weisstein's World of Mathematics, Factorial
Eric Weisstein's World of Mathematics, Left Factorial
G. Xiao, Sigma Server, Operate on "n!"
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.
Index entries for sequences related to factorial numbers

Equals A003422(n+1) - 1.
Cf. A000142, A000670, A001597, A007916, A137593, A143122, A161128, A180632.
Column k=0 of A120695.

List([1..20],n->Sum([1..n],Factorial)); # Muniru A Asiru, Jan 31 2018

a007489 n = a007489_list !! n
a007489_list = scanl (+) 0 $ tail a000142_list
-- Reinhard Zumkeller, Aug 29 2014

[0] cat [&+[Factorial(i): i in [1..n]]: n in [1..25]]; // Vincenzo Librandi, Sep 02 2016

A007489 := proc(n) local i; add(i!,i=1..n); end proc;

FoldList[Plus, 0, (Range@ 21)! ] (* Robert G. Wilson v, Sep 21 2007 *)
Table[Sum[i!, {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
Accumulate[Range[50]!]  (* Harvey P. Dale, Apr 30 2011 *)
Table[Plus@@(Range[n]!), {n, 20}] (* Alonso del Arte, Jul 18 2011 *)

a(n)=sum(k=1,n,k!) \\ Charles R Greathouse IV, Jul 25 2011

a(n) = Sum_{k=1..n} P(n, k)/C(n, k). - Ross La Haye, Sep 21 2004
a(n) = 3*A056199(n) for n>=2. - Philippe Deléham, Feb 10 2007
a(n) = !(n+1)-1=A003422(n+1)-1. - Artur Jasinski, Nov 08 2007 [corrected by Werner Schulte, Oct 20 2021]
Starting (1, 3, 9, 33, 153, ...), = row sums of triangle A137593 - Gary W. Adamson, Jan 28 2008
a(n) = a(n-1) + n! for n >= 1. - Jaroslav Krizek, Jun 16 2009
E.g.f. A(x) satisfies to the differential equation A'(x)=A(x)+x/(1-x)^2+1. - Vladimir Kruchinin, Jan 22 2011
a(0)=0, a(1)=1, a(n) = (n+1)*a(n-1)-n*a(n-2). - Sergei N. Gladkovskii, Jul 05 2012
G.f.: W(0)*x/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
G.f.: x /(1-x)/Q(0),m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
E.g.f.: exp(x-1)*(Ei(1) - Ei(1-x)) - exp(x) + 1/(1 - x), where Ei(x) is the exponential integral. - Ilya Gutkovskiy, Nov 27 2016
a(n) = sqrt(a(n-1)*a(n+1)-a(n-2)*n*n!), n >= 2. - Gary Detlefs, Oct 26 2020
a(n) ~ n!. - Ridouane Oudra, Jun 11 2025

A052410 Write n = m^k with m, k integers, k >= 1, then a(n) is the smallest possible choice for m.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Eric W. Weisstein

nonn

Value of m in m^p = n, where p is the largest possible power (see A052409).
For n > 1, n is a perfect power iff a(n) <> n. - Reinhard Zumkeller, Oct 13 2002
a(n)^A052409(n) = n. - Reinhard Zumkeller, Apr 06 2014
Every integer root of n is a power of a(n). All entries (except 1) belong to A007916. - Gus Wiseman, Sep 11 2017

Daniel Forgues, Table of n, a(n) for n=1..100000
Eric Weisstein's World of Mathematics, Power
Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597, A025478, A007916, A027748, A052409, A072775, A124010, A175781, A278028, A288636, A289023.

a052410 n = product $ zipWith (^)
                      (a027748_row n) (map (`div` (foldl1 gcd es)) es)
            where es = a124010_row n
-- Reinhard Zumkeller, Jul 15 2012

a:= n-> (l-> (t-> mul(i[1]^(i[2]/t), i=l))(
         igcd(seq(i[2], i=l))))(ifactors(n)[2]):
seq(a(n), n=1..74);  # Alois P. Heinz, Jul 22 2024

Table[If[n==1, 1, n^(1/(GCD@@(Last/@FactorInteger[n])))], {n, 100}]

a(n) = if (ispower(n,,&r), r, n); \\ Michel Marcus, Jul 19 2017

def upto(n):
    list = [1] + [0] * (n - 1)
    for i in range(2, n + 1):
        if not list[i - 1]:
            j = i
            while j <= n:
                list[j - 1] = i
                j *= i
    return list
# M. Eren Kesim, Jun 03 2021

from math import gcd
from sympy import integer_nthroot, factorint
def A052410(n): return integer_nthroot(n,gcd(*factorint(n).values()))[0] if n>1 else 1 # Chai Wah Wu, Mar 02 2024

a(A001597(k)) = A025478(k).

a(n) = A007916(A278028(n,1)). - Gus Wiseman, Sep 11 2017

Definition edited (in a complementary form to A052409) by Daniel Forgues, Mar 14 2009
Corrected by Charles R Greathouse IV, Sep 02 2009
Definition edited by N. J. A. Sloane, Sep 03 2010

A052409 a(n) = largest integer power m for which a representation of the form n = k^m exists (for some k).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Eric W. Weisstein

nonn

Greatest common divisor of all prime-exponents in canonical factorization of n for n>1: a(n)>1 iff n is a perfect power; a(A001597(k))=A025479(k). - Reinhard Zumkeller, Oct 13 2002
a(1) set to 0 since there is no largest finite integer power m for which a representation of the form 1 = 1^m exists (infinite largest m). - Daniel Forgues, Mar 06 2009
A052410(n)^a(n) = n. - Reinhard Zumkeller, Apr 06 2014
Positions of 1's are A007916. Smallest base is given by A052410. - Gus Wiseman, Jun 09 2020

			n = 72 = 2*2*2*3*3: GCD[exponents] = GCD[3,2] = 1. This is the least n for which a(n) <> A051904(n), the minimum of exponents.
For n = 10800 = 2^4 * 3^3 * 5^2, GCD[4,3,2] = 1, thus a(10800) = 1.

Daniel Forgues, Table of n, a(n) for n = 1..100000
N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564
Eric Weisstein's World of Mathematics, Power
Eric Weisstein's World of Mathematics, Perfect Power
Index entries for sequences computed from exponents in factorization of n

Cf. A052410, A005361, A051903, A072411-A072414, A124010, A075802, A072776, A270492.
Apart from the initial term essentially the same as A253641.
Differs from A051904 for the first time at n=72, where a(72) = 1, while A051904(72) = 2.
Differs from A158378 for the first time at n=10800, where a(10800) = 1, while A158378(10800) = 2.
Cf. A000005, A000961, A001597, A052410, A303386, A327501.

a052409 1 = 0
a052409 n = foldr1 gcd $ a124010_row n
-- Reinhard Zumkeller, May 26 2012

# See link.
#
a:= n-> igcd(map(i-> i[2], ifactors(n)[2])[]):
seq(a(n), n=1..120);  # Alois P. Heinz, Oct 20 2019

Table[GCD @@ Last /@ FactorInteger[n], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=my(k=ispower(n)); if(k, k, n>1) \\ Charles R Greathouse IV, Oct 30 2012

from math import gcd
from sympy import factorint
def A052409(n): return gcd(*factorint(n).values()) # Chai Wah Wu, Aug 31 2022

(define (A052409 n) (if (= 1 n) 0 (gcd (A067029 n) (A052409 (A028234 n))))) ;; Antti Karttunen, Aug 07 2017

a(1) = 0; for n > 1, a(n) = gcd(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 07 2017

More terms from Labos Elemer, Jun 17 2002

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

cp's OEIS Frontend

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

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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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A000533 a(0)=1; a(n) = 10^n + 1, n >= 1.

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A007916 Numbers that are not perfect powers.

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A025475 1 and the prime powers p^m where m >= 2, thus excluding the primes.

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