A009087 -id:A009087 - cp's OEIS frontend

A141242 a(n) is the number of divisors of the n-th positive integer with a prime number of divisors. In other words, a(n) is the number of divisors of A009087(n).

2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Leroy Quet, Jun 16 2008

nonn

A009087(n) is of the form p^(a(n)-1), where p is some prime.

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

Cf. A000005, A009087, A141241.

DivisorSigma[0, #] &@ Select[Range@ 500, PrimeQ@ DivisorSigma[0, #] &] (* Michael De Vlieger, Aug 19 2017 *)

from sympy import primepi, integer_nthroot, primerange, factorint
def A141242(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k-1)[0]) for k in primerange(x.bit_length()+1)))
    return list(factorint(bisection(f,n,n)).values())[0]+1 # Chai Wah Wu, Feb 22 2025

a(n) = A000005(A009087(n)).

Extended by Ray Chandler, Jun 25 2009

A030629 Numbers with 11 divisors.

Original entry on oeis.org

1024, 59049, 9765625, 282475249, 25937424601, 137858491849, 2015993900449, 6131066257801, 41426511213649, 420707233300201, 819628286980801, 4808584372417849, 13422659310152401

Offset: 1

Jeff Burch

Let p be a prime. Then the n-th number with p divisors is equal to prime(n)^(p-1). - Omar E. Pol, May 06 2008

R. J. Mathar, Table of n, a(n) for n = 1..1000
OEIS Wiki, Index entries for number of divisors

Cf. A000005, A000040, A001248, A008454, A009087, A030514, A030516.

Cf. A013668, A013678.

[p^10: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

(Prime@Range@30)^10  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

is(n)=isprimepower(n)==10 \\ Charles R Greathouse IV, Jun 19 2016

from sympy import prime
def A030629(n): return prime(n)**10 # Chai Wah Wu, Sep 13 2024

[p**10 for p in prime_range(100)]
# Zerinvary Lajos, May 16 2007

a(n) = A000040(n)^10, i.e. tenth power of n-th prime. - Henry Bottomley, Aug 20 2001
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(10)/zeta(20) = 16368226875/(174611*Pi^10) = A013668/A013678.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(10) = 93555/Pi^10 = 1/A013668. (End)

A058061 Number of prime factors (counted with multiplicity) of d(n), the number of divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3

Offset: 1

Labos Elemer, Nov 23 2000

From Bernard Schott, Mar 24 2020: (Start)
a(n) = 1 iff n = p^(q-1) with p, q primes (A009087).
a(n) = 2 if n=p*q with p, q primes (A006881), or if n=p^2*q with p, q primes (A054753), or if n=p^4*q with p, q primes (A178739), or if n=p^6*q with p, q primes (A189987), or if n=p^2*q^4 with p, q primes (A189988), or if n=p^(m-1) with p prime and m is semiprime in A001358 (not exhaustive). (End)

			For n=120, d(120)=16, a(120)=4.

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

Cf. A001222, A000005, A058060, A079057 (partial sums).

Table[PrimeOmega@ DivisorSigma[0, n], {n, 120}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = bigomega(numdiv(n)); \\ Michel Marcus, Dec 14 2013

a(n) = A001222(A000005(n)).

Additive with a(p^e) = A001222(e+1). - Amiram Eldar, Jan 15 2024

A036454 Prime powers with special exponents: q^(p-1) where p > 2 and q are prime numbers.

Original entry on oeis.org

4, 9, 16, 25, 49, 64, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1024, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321

Offset: 1

Labos Elemer

Composite numbers with a prime number of divisors.

			From powers of 2: 4,16,64,1024,4096,65536 are in the sequence since exponent+1 is also prime. The same powers of any prime base are also included.

T. D. Noe, Table of n, a(n) for n = 1..1000

Intersection of A002808 and A009087.

Cf. A000005, A010051, A010553, A010055, A100995, A036450, A036452, A036459, A283262.

a009087 n = a009087_list !! (n-1)
a009087_list = filter ((== 1) . a010051 . (+ 1) . a100995) a000961_list
-- Reinhard Zumkeller, Jun 05 2013

[n: n in [1..20000] | not IsPrime(n) and IsPrime(DivisorSigma(0, n))]; // Vincenzo Librandi, May 19 2015

N:= 10^5:
P1:= select(isprime,[2,seq(2*i+1,i=1..floor((sqrt(N)-1)/2))]):
P2:= select(`<=`,P1,1+ilog2(N))[2..-1]:
S:= {seq(seq(p^(q-1), q = select(`<=`,P2,1+floor(log[p](N)))),p=P1)}:
sort(convert(S,list)); # Robert Israel, May 18 2015

specialPrimePowerQ[n_] := With[{f = FactorInteger[n]}, Length[f] == 1 && PrimeQ[f[[1, 1]]] && f[[1, 2]] > 1 && PrimeQ[f[[1, 2]] + 1]]; Select[Range[20000], specialPrimePowerQ]  (* Jean-François Alcover, Jul 02 2013 *)
Select[Range[20000], ! PrimeQ[#] && PrimeQ[DivisorSigma[0, #]] &] (* Carlos Eduardo Olivieri, May 18 2015 *)

for(n=1,34000, if(isprime(n), n++,x=numdiv(n); if(isprime(x),print(n))))

list(lim)=my(v=List(),t);lim=lim\1+.5;forprime(p=3,log(lim)\log(2) +1, t=p-1; forprime(q=2,lim^(1/t),listput(v,q^t))); vecsort(Vec(v))
\\ Charles R Greathouse IV, Apr 26 2012

from sympy import primepi, integer_nthroot, primerange
def A036454(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p-1)[0]) for p in primerange(3,x.bit_length()+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

d(d(a(n))) = 2, where d(x) = tau(x) = sigma_0(x) is the number of divisors of x.
a(n) = (n log n)^2 + 2n^2 log n log log n + O(n^2 log n). - Charles R Greathouse IV, Apr 26 2012
(1 - A010051(a(n))) * A010055(a(n)) * A010051(A100995(a(n))+1) = 1. - Reinhard Zumkeller, Jun 05 2013
A036459(a(n)) = 2. - Ivan Neretin, Jan 25 2016
a(n) = A283262(n)^2. - Amiram Eldar, Jul 04 2022
Sum_{n>=1} 1/a(n) = Sum_{k>=2} P(prime(k)-1) = 0.54756961912815344341..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

A036436 Numbers whose number of divisors is a square.

Original entry on oeis.org

1, 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR (named after Hardy and Ramanujan) concept formation program.
Numbers in this sequence but not in A036455 are 1, 1260, 1440, 1800, 1980 etc. [From R. J. Mathar, Oct 20 2008]
tau(p^(n^2-1)) = n^2 so numbers of this form are in this sequence, and because tau is multiplicative: if a and b are in this sequence and (a,b)=1 then a*b is also in a(n). - Enrique Pérez Herrero, Jan 22 2013
What is the density of this sequence? It contains A030229 and thus has (lower) density at least 3/Pi^2 = 0.30396...; it does not contain any members of A030059 or A060687, and hence has (upper) density at most 1 - 3/Pi^2 - 6*A179119/Pi^2 = 0.49528.... - Charles R Greathouse IV, Jan 11 2025

			tau(6)=4, which is a square number, so 6 is in this sequence.

S. Colton, Automated Theorem Discovery: A Future Direction for Theorem Provers, 2002.

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics (Unfortunately [403 Forbidden])
S. Colton and A. Bundy, Automatic Concept Formation in Pure Mathematics
S. Colton, A. Bundy and T. Walsh, Agent Based Cooperative Theory Formation in Pure Mathematics
S. Colton, R. McCasland and A. Bundy, Automated Theory Formation for Tutoring Tasks in Pure Mathematics, 2002.

Contains A030229 as a subsequence.

Cf. A009087, A033950, A057265, A049439.

Select[Range[200],IntegerQ[Sqrt[DivisorSigma[0,#]]]&]  (* Harvey P. Dale, Apr 20 2011 *)

is(n)=issquare(numdiv(n)) \\ Charles R Greathouse IV, Jan 22 2013

Links corrected and edited by Daniel Forgues, Jun 30 2010

A063806 Numbers with a prime number of proper divisors.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104

Offset: 1

Henry Bottomley, Aug 20 2001

nonn

			15 has three proper divisors {1,3,5} and so is on the list; 16 has four {1,2,4,8} and so is not; 17 has one {1} and so is not; 18 has five {1,2,3,6,9} and so is.

Harry J. Smith, Table of n, a(n) for n = 1..1000

A subset of A002808. Cf. A009087, A032741.

n=0; for (m=1, 10^9, if(isprime(numdiv(m) - 1), write("b063806.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Aug 31 2009

is(n)=isprime(numdiv(n)-1) \\ Charles R Greathouse IV, Sep 18 2015

A229264 Primes in A065387 in the order of their appearance.

Original entry on oeis.org

2, 19, 19, 79, 103, 113, 257, 523, 509, 1151, 1279, 1193, 1579, 2273, 3061, 2389, 2693, 2843, 5003, 4831, 5119, 7411, 5693, 5623, 8623, 6323, 10139, 8933, 18401, 14957, 20411, 20479, 21191, 20123, 29683, 28211, 36833, 55021, 57203, 68743, 48761, 66533, 62423

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			Third term of A038344 is 9 and sigma(9) + phi(9) = 13 + 6 = 19 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065387, A115919, A141242, A229265, A229266, A229267, A229268.

with(numtheory); P:=proc(q) local a, n; for n from 1 to q do a:=sigma(n)+phi(n);
if isprime(a) then print(a); fi; od; end: P(10^6);

Select[Table[DivisorSigma[1,n]+EulerPhi[n],{n,30000}],PrimeQ] (* Harvey P. Dale, Apr 30 2018 *)

lista(kmax) = {my(f, s); for(k = 1, kmax, f = factor(k); s= sigma(f) + eulerphi(f); if(isprime(s), print1(s, ", ")));} \\ Amiram Eldar, Nov 19 2024

Name corrected by Amiram Eldar, Nov 19 2024

A229268 Primes of the form sigma(k) - tau(k), where sigma(k) = A000203(k) and tau(k) = A000005(k).

Original entry on oeis.org

2, 11, 353, 1013, 2333, 16369, 58579, 65519, 123733, 1982273, 7089683, 5778653, 12795053, 10500593, 22586027, 19980143, 24126653, 67108837, 72494713, 90781993, 106199593, 203275951, 164118923, 183105421, 320210549, 259997173, 794091653, 1279963973

Offset: 1

Paolo P. Lava, Sep 18 2013

nonn

			Second term of A065061 is 8 and sigma(8) - tau(8) = 15 - 4 = 11 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000005, A000010, A000203, A009087, A023194, A038344, A055813, A062700, A064205, A065608, A141242, A229264, A229266

with(numtheory); P:=proc(q) local a,n; a:= sigma(n)-tau(n); for n from 1 to q do
if isprime(a) then print(a); fi; od; end: P(10^6);

Join[{2}, Select[(DivisorSigma[1, #] - DivisorSigma[0, #]) & /@ (2*Range[20000]^2), PrimeQ]] (* Amiram Eldar, Dec 06 2022 *)

a(n) = A000203(A065061(n)) - A000005(A065061(n)). - Michel Marcus, Sep 21 2013

a(n) = A065608(A065061(n)). - Amiram Eldar, Dec 06 2022

More terms from Michel Marcus, Sep 21 2013

A139118 Numbers with a nonprime number of divisors.

Original entry on oeis.org

1, 6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 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

Offset: 1

Omar E. Pol, May 09 2008

A000005(a(n)) is nonprime. Complement of A009087. Also, nonprime numbers with nonprime number of divisors.

The sequence consists of those n such that n is not a prime power, or n of the form p^k where k+1 is composite. - Franklin T. Adams-Watters, Apr 09 2009

			15 is in the sequence because it has 4 divisors: 1, 3, 5, and 15. - _Emeric Deutsch_, Jun 27 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A009087, A018252.

with(numtheory): a := proc (n) if isprime(tau(n)) = false then n else end if end proc: seq(a(n), n = 1 .. 120); # Emeric Deutsch, Jun 27 2009

Select[Range[102], ! PrimeQ[DivisorSigma[0, #]] &] (* Amiram Eldar, Nov 27 2020 *)

is(n)=!isprime(numdiv(n)) \\ Charles R Greathouse IV, Jun 19 2016

from sympy import primepi, integer_nthroot, primerange
def A139118(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k-1)[0]) for k in primerange(x.bit_length()+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 22 2025

Extended by Ray Chandler, Jun 25 2009

A203967 The number of positive integers <= n that have a prime number of divisors.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25

Offset: 1

Geoffrey Critzer, Jan 08 2012

nonn

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

Cf. A009087, A010051, A010055, A100995.

a203967 n = length $ takeWhile (<= n) a009087_list
-- Reinhard Zumkeller, Jun 06 2013

Table[Total[Table[PrimePi[m^(1/n)], {n,Table[Prime[n]-1, {n,1,20}]}]], {m,1,100}]
tot = 0; Table[If[PrimeQ[DivisorSigma[0, n]], tot++]; tot, {n, 100}] (* T. D. Noe, Jan 10 2012 *)

from sympy import primepi, integer_nthroot, primerange
def A203967(n): return int(sum(primepi(integer_nthroot(n,k-1)[0]) for k in primerange(n.bit_length()+1))) # Chai Wah Wu, Feb 22 2025

a(n) = a(n-1) + A010055(n) * A010051(A100995(n)+1). - Reinhard Zumkeller, Jun 06 2013

cp's OEIS Frontend

A141242 a(n) is the number of divisors of the n-th positive integer with a prime number of divisors. In other words, a(n) is the number of divisors of A009087(n).

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A030629 Numbers with 11 divisors.

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A058061 Number of prime factors (counted with multiplicity) of d(n), the number of divisors of n.

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A036454 Prime powers with special exponents: q^(p-1) where p > 2 and q are prime numbers.

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A036436 Numbers whose number of divisors is a square.

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A063806 Numbers with a prime number of proper divisors.

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A229264 Primes in A065387 in the order of their appearance.