A010055 -id:A010055 - 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

A024619 Numbers that are not powers of primes p^k (k >= 0); complement of A000961.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Clark Kimberling

The sequence of numbers divisible by a prime number of primes coincides with this up to 210, which has 4 prime factors. - Lior Manor, Aug 23 2001
A085970(n) = Max{k: a(k)<=n}.
Numbers n such that LCM of proper divisors of n equals neither 1 nor n. - Labos Elemer, Dec 01 2004
a(n) provides bases b in which automorphic numbers m^2 ending with m in base b exist. In the complement there aren't any automorphic numbers. - Martin Renner, Dec 07 2011
Numbers with at least 2 distinct prime factors. - Jonathan Sondow, Oct 17 2013
There exists an equiangular n-gon whose edge lengths form a permutation of 1, 2, ..., n if and only if n is in the sequence (see Woeginger's survey and Munteanu & Munteanu). - Jonathan Sondow, Oct 17 2013
Numbers that are the product of two relatively prime factors. These numbers are used in testing a sequence for multiplicativity. - Michael Somos, Jun 02 2015
A theorem from Donald McCarthy: Let d be any positive integer which is not a prime power; then there exists a finite group whose order is divisible by d but which contains no subgroup of order d (see link and A340511). - Bernard Schott, Dec 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 8719 terms from Daniel Forgues)
Donald McCarthy, Sylow's theorem is a sharp partial converse to Lagrange's theorem, Mathematische Zeitschrift, 113, 383-384 (1970).
Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 1460-1465.
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
Wikipedia, Prime power
G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.
Günter Ziegler and Brady Haran, Cannons and Sparrows, Numberphile video (2018).

Cf. A000040, A000961 (complement), A001221, A014963, A020500, A085970.
Cf. A340511.
Subsequence of A080257.

a024619 n = a024619_list !! (n-1)
a024619_list = filter ((== 0) . a010055) [1..]
-- Reinhard Zumkeller, Nov 17 2011

IsA024619:=func< n | not IsPrime(n) and not (t and IsPrime(b) where t, b, A024619(n)%20%5D;%20//%20_Klaus%20Brockhaus">:=IsPower(n)) >; [ n: n in [2..200] | IsA024619(n) ]; // _Klaus Brockhaus, Feb 25 2011

a := proc(n) numtheory[factorset](n); if 1 < nops(%) then n else NULL fi end:
seq(a(i), i=1..110); # Peter Luschny, Aug 11 2009

Select[Range@111, Length@FactorInteger@# > 1 &] (* Robert G. Wilson v, Dec 07 2005 *)

is(n)=n>5 && !isprimepower(n) \\ Charles R Greathouse IV, Mar 21 2013

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A024619(n):
    def f(x): return int(n+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 A024619_list(n) :
    return [k for k in (2..n) if not k.is_prime() and not k.is_prime_power()]
A024619_list(112)  # Peter Luschny, Feb 03 2012 [corrected by Terry D. Grant, Sep 16 2020]

A001221(a(n)) > 1.
A014963(a(n)) = 1.
A020500(a(n)) = 1. - Benoit Cloitre, Aug 26 2003
A010055(a(n)) = 0. - Reinhard Zumkeller, Nov 17 2011
a(n) ~ n. - Charles R Greathouse IV, Mar 21 2013
a(n) ~ n - pi(n) [See Panaitopol]. - N. J. A. Sloane, Sep 27 2020
A118887(a(n)) > 0. - Jonathan Sondow, Oct 17 2013

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

A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1

Offset: 1

Marc LeBrun

There are arbitrarily long runs of ones (Sierpiński). - Franz Vrabec, Sep 26 2005
a(n) is the smallest positive integer such that n divides Product_{k=1..n} a(k), for all positive integers n. - Leroy Quet, May 01 2007
For n>1, resultant of the n-th cyclotomic polynomial with the 1st cyclotomic polynomial x-1. - Ralf Stephan, Aug 14 2013
A368749(n) is the smallest prime p such that the interval [a(p), a(q)] contains n 1's; q = nextprime(p), n >= 0. - David James Sycamore, Mar 21 2024

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 17.7.
I. Vardi, Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 146-147, 152-153 and 249, 1991.

T. D. Noe, Table of n, a(n) for n = 1..10000
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384 [math.CA], 2009.
Carl McTague, On the Greatest Common Divisor of C(q*n,n), C(q*n,2*n), ...C(q*n,q*n-q), arXiv:1510.06696 [math.CO], 2015.
A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.
W. Sierpiński, On the numbers [1,2,...n], (Polish) Wiadom. Mat. (2) 9 1966 9-10.
Eric Weisstein's World of Mathematics, Mangoldt Function.
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.
Index entries for sequences related to lcm's

Apart from initial 1, same as A020500. With ones replaced by zeros, equal to A120007.
Cf. A003418, A007947, A008683, A008472, A008578, A048671 (= n/a(n)), A072107 (partial sums), A081386, A081387, A099636, A100994, A100995, A140255 (inverse Mobius transform), A140254 (Mobius transform), A297108, A297109, A340675, A000027, A348846, A368749.
First column of A140256. Row sums of triangle A140581.
Cf. also A140579, A140580 (= n*a(n)).

a014963 1 = 1
a014963 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
          | otherwise = 1
          where spf = a020639 n
-- Reinhard Zumkeller, Sep 09 2011

a := n -> if n < 2 then 1 else numtheory[factorset](n); if 1 < nops(%) then 1 else op(%) fi fi; # Peter Luschny, Jun 23 2009
A014963 := n -> n/ilcm(op(numtheory[divisors](n) minus {1,n}));
seq(A014963(i), i=1..69); # Peter Luschny, Mar 23 2011
# The following is Nowicki's LCM-Transform - N. J. A. Sloane, Jan 09 2024
LCMXFM:=proc(a)  local p,q,b,i,k,n:
if whattype(a) <> list then RETURN([]); fi:
n:=nops(a):
b:=[a[1]]: p:=[a[1]];
for i from 2 to n do q:=[op(p),a[i]]; k := lcm(op(q))/lcm(op(p));
b:=[op(b),k]; p:=q;; od:
RETURN(b); end:
# Alternative, to be called by 'seq' as shown, not for a single n.
a := proc(n) option remember; local i; global f; f := ifelse(n=1, 1, f*n);
iquo(f, mul(a(i)^iquo(n, i), i=1..n-1)) end: seq(a(n), n=1..95); # Peter Luschny, Apr 05 2025

a[n_?PrimeQ] := n; a[n_/;Length[FactorInteger[n]] == 1] := FactorInteger[n][[1]][[1]]; a[n_] := 1; Table[a[n], {n, 95}] (* Alonso del Arte, Jan 16 2011 *)
a[n_] := Exp[ MangoldtLambda[n]]; Table[a[n], {n, 95}] (* Jean-François Alcover, Jul 29 2013 *)
Ratios[LCM @@ # & /@ Table[Range[n], {n, 100}]] (* Horst H. Manninger, Mar 08 2024 *)
Table[Which[PrimeQ[n],n,PrimePowerQ[n],Surd[n,FactorInteger[n][[-1,2]]],True,1],{n,100}] (* Harvey P. Dale, Mar 01 2025 *)

A014963(n)=
{
    local(r);
    if( isprime(n), return(n));
    if( ispower(n,,&r) && isprime(r), return(r) );
    return(1);
}  \\ Joerg Arndt, Jan 16 2011

a(n)=ispower(n,,&n);if(isprime(n),n,1) \\ Charles R Greathouse IV, Jun 10 2011

from sympy import factorint
def A014963(n):
    y = factorint(n)
    return list(y.keys())[0] if len(y) == 1 else 1
print([A014963(n) for n in range(1, 71)]) # Chai Wah Wu, Sep 04 2014

def A014963(n) : return simplify(exp(add(moebius(d)*log(n/d) for d in divisors(n))))
[A014963(n) for n in (1..50)]  # Peter Luschny, Feb 02 2012

def a(n):
    if n == 1: return 1
    return prod(1 - E(n)**k for k in ZZ(n).coprime_integers(n+1))
[a(n) for n in range(1, 14)] # F. Chapoton, Mar 17 2020

a(n) = A003418(n) / A003418(n-1) = lcm {1..n} / lcm {1..n-1}. [This is equivalent to saying that this sequence is the LCM-transform (as defined by Nowicki, 2013) of the positive integers. - David James Sycamore, Jan 09 2024.]
a(n) = 1/Product_{d|n} d^mu(d) = Product_{d|n} (n/d)^mu(d). - Vladeta Jovovic, Jan 24 2002
a(n) = gcd( C(n+1,1), C(n+2,2), ..., C(2n,n) ) where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003
a(n) = gcd(C(n,1), C(n+1,2), C(n+2,3), ...., C(2n-2,n-1)), where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003; corrected by Ant King, Dec 27 2005
Note: a(n) != gcd(A008472(n), A007947(n)) = A099636(n), GCD of rad(n) and sopf(n) (this fails for the first time at n=30), since a(30) = 1 but gcd(rad(30), sopf(30)) = gcd(30,10) = 10.
a(n)^A100995(n) = A100994(n). - N. J. A. Sloane, Feb 20 2005
a(n) = Product_{k=1..n-1, if(gcd(n, k)=1, 1-exp(2*Pi*i*k/n), 1)}, i=sqrt(-1); a(n) = n/A048671(n). - Paul Barry, Apr 15 2005
Sum_{n>=1} (log(a(n))-1)/n = -2*A001620 [Bateman Manuscript Project Vol III, ed. by Erdelyi et al.]. - R. J. Mathar, Mar 09 2008
n*a(n) = A140580(n) = n^2/A048671(n) = A140579 * [1,2,3,...]. - Gary W. Adamson, May 17 2008
a(n) = (2*Pi)^phi(n) / Product_{gcd(n,k)=1} Gamma(k/n)^2 (for n > 1). - Peter Luschny, Aug 08 2009
a(n) = A166140(n) / A166142(n). - Mats Granvik, Oct 08 2009
a(n) = GCD of rows in A167990. - Mats Granvik, Nov 16 2009
a(n) = A010055(n)*(A007947(n) - 1) + 1. - Reinhard Zumkeller, Mar 26 2010
a(n) = 1 + (A007947(n)-1) * floor(1/A001221(n)), for n>1. - Enrique Pérez Herrero, Jun 01 2011
a(n) = Product_{k=1..n-1} if(gcd(k,n)=1, 2*sin(Pi*k/n), 1). - Peter Luschny, Jun 09 2011
a(n) = exp(Sum_{k>=1} A191898(n,k)/k) for n>1 (conjecture). - Mats Granvik, Jun 19 2011
Dirichlet g.f.: Sum_{n>0} e^Lambda(n)/n^s = Zeta(s) + Sum_{p prime} Sum_{k>0} (p-1)/p^(k*s) = Zeta(s) - ppzeta(s) + Sum(p prime, p/(p^s-1)); for a ppzeta definition see A010055. - Enrique Pérez Herrero, Jan 19 2013
a(n) = exp(lim_{s->1} zeta(s)*Sum_{d|n} moebius(d)/d^(s-1)) for n>1. - Mats Granvik, Jul 31 2013
a(n) = gcd_{k=1..n-1} binomial(n,k) for n > 1, see A014410. - Michel Marcus, Dec 08 2015 [Corrected by Jinyuan Wang, Mar 20 2020]
a(n) = 1 + Sum_{k=2..n} (k-1)*A010051(k)*(floor(k^n/n) - floor((k^n - 1)/n)). - Anthony Browne, Jun 16 2016
The Dirichlet series for log(a(n)) = Lambda(n) is given by the logarithmic derivative of the zeta function -zeta'(s)/zeta(s). - Mats Granvik, Oct 30 2016
a(n) = A008578(1+A297109(n)), For all n >= 1, Product_{d|n} a(d) = n. - Antti Karttunen, Feb 01 2021
Product_{k=1..floor(n/2)} Product_{j=1..floor(n/k)} a(j) = n!. - Ammar Khatab, Jan 28 2025

Additional reference from Eric W. Weisstein, Jun 29 2008

A034699 Largest prime power factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79

Offset: 1

David W. Wilson

n divides lcm(1, 2, ..., a(n)).
a(n) = A210208(n,A073093(n)) = largest term of n-th row in A210208. - Reinhard Zumkeller, Mar 18 2012
a(n) = smallest m > 0 such that n divides A003418(m). - Thomas Ordowski, Nov 15 2013
a(n) = n when n is a prime power (A000961). - Michel Marcus, Dec 03 2013
Conjecture: For all n between two consecutive prime numbers, all a(n) are different. - I. V. Serov, Jun 19 2019
Disproved with between p=prime(574) = 4177 and prime(575) = 4201, a(4180) = a(4199) = 19. See A308752. - Michel Marcus, Jun 19 2019
Conjecture: For any N > 0, there exist numbers n and m, N < n < n+a(n) <= m, such that all n..m are composite and a(n) = a(m). - I. V. Serov, Jun 21 2019
Conjecture: For all n between two consecutive prime numbers, all (-1)^n*a(n) are different. Checked up to 5*10^7. - I. V. Serov, Jun 23 2019
Disproved: between p = prime(460269635) = 10120168277 and p = prime(460269636) = 10120168507 the numbers n = 10120168284 and m = 10120168498 form a pair such that (-1)^n*a(n) = (-1)^m*a(m) = 107. - L. Joris Perrenet, Jan 05 2020
a(n) = cardinality of smallest set on which idempotence of order n+1 (f^{n+1} = f) differs from idempotence of order e for 2 <= e <= n (see von Eitzen link for proof); derivable from A245501. - Mark Bowron, May 22 2025

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
Hagen von Eitzen, Least k to distinguish f^n = f on {1,...,k} from f^e = f for every 1 < e < n is largest prime power factor of n-1?, Math Stack Exchange.

Cf. A006530, A010055, A020639, A027750, A034684, A028233, A051283, A052128, A053585, A057110, A060818, A038610, A081805, A088387, A088388, A100574, A210208, A245501, A284600, A308752.

Numbers n ordered by a(n): A305325.

a034699 = last . a210208_row
-- Reinhard Zumkeller, Mar 18 2012, Feb 14 2012

f[n_] := If[n == 1, 1, Max[ #[[1]]^#[[2]] & /@ FactorInteger@n]]; Array[f, 79] (* Robert G. Wilson v, Sep 02 2006 *)
Array[Max[Power @@@ FactorInteger@ #] &, 79] (* Michael De Vlieger, Jul 26 2018 *)

a(n) = if(1==n,n,my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))); \\ Charles R Greathouse IV, Nov 20 2012, check for a(1) added by Antti Karttunen, Aug 06 2018

A034699(n) = if(1==n,n,fordiv(n, d, if(isprimepower(n/d), return(n/d)))); \\ Antti Karttunen, Aug 06 2018

from sympy import factorint
def A034699(n): return max((p**e for p, e in factorint(n).items()), default=1) # Chai Wah Wu, Apr 17 2023

If n = p_1^e_1 *...* p_k^e_k, p_1 < ... < p_k primes, then a(n) = Max_i p_i^e_i.
a(n) = A088387(n)^A088388(n). - Antti Karttunen, Jul 22 2018
a(n) = n/A284600(n) = n - A081805(n) = A034684(n) + A100574(n). - Antti Karttunen, Aug 06 2018
a(n) = a(m) iff m = d*a(n), where d is a divisor of A038610(a(n)). - I. V. Serov, Jun 19 2019

A006549 Numbers k such that k and k+1 are prime powers.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 16, 31, 127, 256, 8191, 65536, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

N. J. A. Sloane

Numbers k such that k + (0, 1) is a prime power pair.
Consecutive prime powers.
k + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) == (0, 0) (mod 2), has high density.
k + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) == (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0].
Numbers k such that k and k+1 are primes would give only 2, for the prime pair (2, 3).
This sequence corresponds to the least member of each one of the following prime power pairs, ordered by increasing value of least member: (1, 2), (2^3, 3^2), (Fermat primes - 1, Fermat primes), (Mersenne primes, Mersenne primes + 1).
It is not known whether this sequence is infinite, but is conjectured to be since:
(*) 2^3, 3^2 are the only consecutive prime powers with exponents >= 2
(as a consequence of Mihailescu's theorem -- Mihailescu proved Catalan's conjecture in 2002);
(*) Only the first 5 Fermat numbers f_0 to f_4 are known to be prime
(it is conjectured that there might be no others, f_5 to f_32 are all composite);
(*) It has been conjectured that there exist an infinite number of Mersenne primes.
Numbers k such that A003418(k) appears only once in the sequence A003418. This may suggest that k is also characterized by the pairs formed by a 2 whose direct neighbor is a prime number in the sequence A014963. - Eric Desbiaux, Feb 11 2015
The power graph and enhanced power graph of the groups PGL(2,q) have the same clique number iff q>1 is a term of this sequence (Peter Cameron's link). - Bernard Schott, Dec 14 2021

R. K. Guy, Unsolved Problems in Number Theory, D9.
P. Ribenboim, 13 Lect. on Fermat's Last Theorem, p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David W. Wilson and Eric Rains (rains(AT)caltech.edu) found a simple proof that in this case of Catalan's conjecture either n or n+1 must be a power of 2 and the other number must be a prime, except for n=8. Using this the sequence is easy to extend.

Daniel Forgues, Table of n, a(n) for n = 1..25
Peter Cameron, Graphs on groups, 9, Peter Cameron's blog.
Wacław Sierpiński, Sur une question concernant le nombre de diviseurs premiers d'un nombre naturel, Colloquium Mathematicum 6 (1958), 209-210.
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Eric Weisstein's World of Mathematics, Clique Number.
Eric Weisstein's World of Mathematics, Fermat Prime.
Eric Weisstein's World of Mathematics, Mersenne Prime.

Cf. A000961, A000040, A010055.
Cf. A019434 Fermat primes: primes of form 2^(2^n) + 1, n >= 0.
Cf. A000668 Mersenne primes (of form 2^p - 1 where p is a prime).
Cf. A120431 Numbers n such that n and n+2 are prime powers.
Cf. A164571 Numbers n such that n and n+3 are prime powers.
Cf. A164572 Numbers n such that n and n+4 are prime powers.
Cf. A164573 Numbers n such that n and n+5 are prime powers.
Cf. A164574 Numbers n such that n and n+6 are prime powers.

a006549 n = a006549_list !! (n-1)
a006549_list = [1,2,3,4,7,8] ++ f (drop 4 a000040_list) where
   f (p:ps) | a010055 (p - 1) == 1 = (p - 1) : f ps
            | a010055 (p + 1) == 1 = p : f ps
            | otherwise            = f ps
-- Reinhard Zumkeller, Jan 03 2013

Do[ a = Length[ FactorInteger[ 2^n - 1 ] ]; b = Length[ FactorInteger[ 2^n ] ]; c = Length[ FactorInteger[ 2^n + 1 ] ]; If[ a == b, Print[ 2^n - 1 ] ]; If[ b == c, Print[ 2^n ] ], {n, 0, 127} ]
Join[{1},SequencePosition[Boole[PrimePowerQ[Range[600000]]],{1,1}][[All,1]]] (* Requires Mathematica version 10 or later *) (* Generates the first 14 terms of the sequence. Increase Range constant to generate more. *) (* Harvey P. Dale, Apr 12 2020 *)

is(n)=if(n<5,return(n>0)); isprimepower(n) && isprimepower(n+1) \\ Charles R Greathouse IV, Apr 24 2015

More terms from David W. Wilson

Additional comments from Daniel Forgues, Aug 17 2009

A053810 Numbers of the form p^e where both p and e are prime numbers.

Original entry on oeis.org

4, 8, 9, 25, 27, 32, 49, 121, 125, 128, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 8192, 9409, 10201, 10609, 11449, 11881, 12167

Offset: 1

Henry Bottomley, Mar 28 2000

Possible orders of finite fields with exactly 2 subfields. In other words, possible orders of finite fields whose only proper subfield is the prime field. - Jianing Song, Jun 06 2025

T. D. Noe, Table of n, a(n) for n = 1..9965

Cf. A000040, A000961, A053811, A053812.
Cf. A203967; subsequence of A000961.
Cf. A113877 (similar for semiprimes).

a053810 n = a053810_list !! (n-1)
a053810_list = filter ((== 1) . a010051 . a100995) $ tail a000961_list
-- Reinhard Zumkeller, Jun 05 2013

h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 1 and isprime(padic:-ordp(n, P[1])) end:
A053810List := upto -> seq(n, n = select(h, [seq(1..upto)])):  # Peter Luschny, Apr 14 2025

pp={}; Do[if=FactorInteger[n]; If[Length[if]==1&&PrimeQ[if[[1, 1]]]&&PrimeQ[if[[1, 2]]], pp=Append[pp, n]], {n, 13000}]; pp
Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[12800]]}, {i, 1, PrimePi[ Log[ Prime[n], 12800]]}]]]

is(n)=isprime(isprimepower(n)) \\ Charles R Greathouse IV, Mar 19 2013

from sympy import primepi, integer_nthroot, primerange
def A053810(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(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 isA(n):
    p = prime_divisors(n)
    return len(p) == 1 and is_prime(valuation(n, p[0]))
print([n for n in srange(1, 12222) if isA(n)])  # Peter Luschny, Apr 14 2025

a(n) = A053811(n)^A053812(n). - David Wasserman, Feb 17 2006
A010055(a(n)) * A010051(A100995(a(n))) = 1. - Reinhard Zumkeller, Jun 05 2013
Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) = 0.6716752222..., where P is the prime zeta function. - Amiram Eldar, Nov 21 2020

More terms from David Wasserman, Feb 17 2006

Name clarified by Peter Luschny, Apr 14 2025

A000430 Primes and squares of primes.

Original entry on oeis.org

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

Offset: 1

R. Muller

Also numbers n such that the product of proper divisors is < n.
See A050216 for lengths of blocks of consecutive primes. - Reinhard Zumkeller, Sep 23 2011
Numbers q > 1 such that d(q) < 4. Numbers k such that the number of ways of writing k = m + t is equal to the number of ways of writing k = r*s, where m|t and r|s. - Juri-Stepan Gerasimov, Oct 14 2017
Called multiplicatively deficient numbers by Chau (2004). - Amiram Eldar, Jun 29 2022

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

T. D. Noe, Table of n, a(n) for n = 1..1000
William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Union of A000040 and A001248.

Cf. A007422, A010051, A010055, A032741, A046951, A050216, A056595, A058080, A064911, A084110, A084114, A293575.

a000430 n = a000430_list !! (n-1)
a000430_list = m a000040_list a001248_list where
   m (x:xs) (y:ys) | x < y = x : m xs (y:ys)
                   | x > y = y : m (x:xs) ys
-- Reinhard Zumkeller, Sep 23 2011

nn = 223; t = Union[Prime[Range[PrimePi[nn]]], Prime[Range[PrimePi[Sqrt[nn]]]]^2] (* T. D. Noe, Apr 11 2011 *)
Module[{upto=250,prs},prs=Prime[Range[PrimePi[upto]]];Select[Join[ prs,prs^2], #<=upto&]]//Sort (* Harvey P. Dale, Oct 08 2016 *)

is(n)=isprime(n) || (issquare(n,&n) && isprime(n)) \\ Charles R Greathouse IV, Sep 04 2013

from math import isqrt
from sympy import primepi
def A000430(n):
    def f(x): return n+x-primepi(x)-primepi(isqrt(x))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

A084114(a(n)) = 0, see also A084110. - Reinhard Zumkeller, May 12 2003
A109810(a(n)) = 2. - Reinhard Zumkeller, May 24 2010
A010051(a(n)) + A010055(a(n))*A064911(a(n)) = 1;
A056595(a(n)) = 1. - Reinhard Zumkeller, Aug 15 2011
A032741(a(n)) = A046951(a(n)); A293575(a(n)) = 0. - Juri-Stepan Gerasimov, Oct 14 2017
The number of terms not exceeding x is N(x) ~ (x + 2*sqrt(x))/log(x) (Chau, 2004). - Amiram Eldar, Jun 29 2022

A073093 Number of prime power divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 3, 3, 2, 5, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 3, 5, 2, 3, 3, 5, 2, 4, 2, 4, 4, 3, 2, 6, 3, 4, 3, 4, 2, 5, 3, 5, 3, 3, 2, 5, 2, 3, 4, 7, 3, 4, 2, 4, 3, 4, 2, 6, 2, 3, 4, 4, 3, 4, 2, 6, 5, 3, 2, 5, 3, 3, 3, 5, 2, 5, 3, 4, 3, 3, 3, 7, 2, 4, 4, 5, 2, 4, 2, 5, 4

Offset: 1

Reinhard Zumkeller, Aug 24 2002

Also, number of prime divisors of 2n (counted with multiplicity).
A001221(n) < a(n) <= A000005(n) for all n; a(n)=A001221(n)+1 iff n is squarefree (A005117); a(n)=A000005(n) iff n is a prime power (A000961).
a(n) is also the number of kBenoit Cloitre, Oct 13 2002
a(n) is also 1 + the number of divisors of n with omega(d)=1, where omega is A001221. - Enrique Pérez Herrero, Nov 05 2009
Length of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012
a(n) depends only on the prime signature of n with a(A025487(n)) = 1, 2, 3, 3, 4, 4, 5, 5, 4, 6, 5, 6, 5, 7, 6, 7 ,.. = A036041(n)+1; (n>=1). - R. J. Mathar, May 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970.
T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A000961, A023888, A054372. Bisection of A001222.

a073093 = length . a210208_row  -- Reinhard Zumkeller, Mar 18 2012

[n eq 1 select 1 else &+[p[2]: p in Factorization(n)]+1: n in [1..100]]; // Vincenzo Librandi, Jan 06 2017

seq(numtheory:-bigomega(n)+1, n=1..1000); # Robert Israel, Sep 06 2015

f[n_] := Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; Table[ f[2n], {n, 105}] (* Robert G. Wilson v, Dec 23 2004 *)
A001221[n_] := (Length[ FactorInteger[n]]); SetAttributes[A001221, Listable]; A073093[n_]:=Length[Select[A001221[Divisors[n]], # == 1 &]]; (* Enrique Pérez Herrero, Nov 05 2009 *)
PrimeOmega[Range[100]] + 1 (* Paolo Xausa, Nov 23 2024 *)

numlib::Omega (2*n)$ n=1..105 // Zerinvary Lajos, May 13 2008

a(n)=sum(k=1,n,if(1-polresultant(polcyclo(n),polcyclo(k)),1,0))

A073093(n)=bigomega(n)+1   \\ M. F. Hasler, Dec 08 2010

If n = Product (p_j^k_j), a(n) = 1 + Sum (k_j).
a(n) = bigomega(n)+1 = A001222(n)+1 = A001222(2*n).
a(n) = if n=1 then 1 else a(A032742(n)) + 1. - Reinhard Zumkeller, Sep 24 2009
a(n) = max { a(d) ; d 1. - David W. Wilson, Dec 08 2010
a(n) = Sum_{k = 1 .. A001221(n)} A010055(A027750(n,k)). - Reinhard Zumkeller, Mar 18 2012
G.f.: x/(1 - x) + Sum_{k>=2} floor(1/omega(k))*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017

cp's OEIS Frontend

A355817 Dirichlet inverse of A010055, characteristic function of powers of primes.

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

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A024619 Numbers that are not powers of primes p^k (k >= 0); complement of A000961.

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

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A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.

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