A010553 -id:A010553 - cp's OEIS frontend

A335831 Numbers k with a record value of tau(tau(k)) (A010553), where tau(k) is the number of divisors of k (A000005).

1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 129729600, 908107200, 2205403200, 15437822400, 293318625600, 3226504881600, 6746328388800, 74209612276800, 195643523275200, 1855240306920000, 2152078756027200, 27977023828353600

Offset: 1

Amiram Eldar, Jun 25 2020

nonn

First differs from A189394 at n=15.

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..73
Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford and Jan-Christoph Schlage-Puchta, A problem of Ramanujan, Erdős, and Kátai on the iterated divisor function, International Mathematics Research Notices, Vol. 2012, No. 17 (2012), pp. 4051-4061, preprint, arXiv:1108.1815 [math.NT], 2011.
Amiram Eldar, Table of n, a(n), A010553(a(n)) for n = 1..73
Christian Elsholtz, Marc Technau and Niclas Technau, The maximal order of iterated multiplicative functions, Mathematika, Vol. 65, No. 4 (2019), pp. 990-1009, preprint, arXiv:1709.04799 [math.NT], 2017 and 2019.

Subsequence of A025487.

Cf. A000005, A005179, A010553, A189394, A193987.

f[n_] := DivisorSigma[0, DivisorSigma[0, n]]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^5}]; s

tau(tau(a(n))) ~ c * sqrt(log(a(n)))/log(log(a(n))), where c is a constant (Buttkewitz et al., 2012).

A083399 Number of divisors of n that are not divisors of other divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4

Offset: 1

Reinhard Zumkeller, Jun 12 2003

a(n) <= tau(n); a(n) = tau(n) iff n is prime or n=1 (A008578, A000040); a(n)=tau(n)-1 iff n is semiprime (A001358).
Number of noncomposite divisors of n, (cf. A008578). - Jaroslav Krizek, Nov 25 2009
From Wilf A. Wilson, Jul 21 2017: (Start)
a(n) is the number of maximal subsemigroups of the annular Jones monoid of degree n.
a(n) is the number of maximal subsemigroups of the monoid of orientation-preserving mappings on a set with n elements.
a(n) + 1 is the number of maximal subsemigroups of the monoid of orientation-preserving partial mappings on a set with n elements.
(End)
This is the restricted growth sequence transform of A001221 (and thus also of A007875, A034444, A082476, A292586 and many other sequences). This follows from the formula a(n) = 1+A001221(n), and from the fact that for any n, A001221(n) <= 1+A001221(k) for all k = 1..(n-1). A067003 gives the ordinal transform of A001221. See also A292582, A292583, A292585. - Antti Karttunen, Sep 25 2017

			{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2, 3, 4 and 6 divide not only 24, but also 8 or 12, therefore a(24) = 3.
{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2 and 3 are noncomposites, therefore a(24) = 3. - _Jaroslav Krizek_, Nov 25 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]

Cf. A000005, A001221, A067003, A055212, A077761.

a083399 = (+ 1) . a001221  -- Reinhard Zumkeller, Sep 14 2015

[(#(PrimeDivisors(n)))+1: n in [1..100]]; // Vincenzo Librandi, Feb 15 2015

A083399 := proc(n)
    1+nops(numtheory[factorset](n)) ;
end proc:
seq(A083399(n),n=1..100) ; # R. J. Mathar, Sep 26 2017

A083399[n_Integer]:=1+PrimeNu[n]; (* Enrique Pérez Herrero, Jun 25 2010 *)
Rest@ CoefficientList[ Series[(1/(1 - x)) + Sum[1/(1 - x^Prime[j]), {j, 200}], {x, 0, 111}], x] (* Robert G. Wilson v, Aug 16 2011 *)

a(n)=#factor(n)~+1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = omega(n) + 1, where omega = A001221.
a(n) = tau(n) - A055212(n) = A000005(n)-A055212(n).
a(n) = A000005(n) - A033273(n) + 1. - Jaroslav Krizek, Nov 25 2009
a(n) = A010553(A007947(n)) = A000005(A000005(A007947(n))) = tau_2(tau_2(rad(n))). - Enrique Pérez Herrero, Jun 25 2010
G.f.: x/(1 - x) + Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) = n * (log(log(n)) + B + 1) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 29 2024

A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

Original entry on oeis.org

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

Offset: 2

Matthew Vandermast, Jun 03 2012

Length of row n equals A001221(n).
The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences.  Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.
(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)
Table lists exponents in the order in which they appear in the prime factorization of a member of A025487.  This ordering is common in database comments (e.g., A008966).
Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.
Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022

			First rows of table read:
  1;
  1;
  2;
  1;
  1,1;
  1;
  3;
  2;
  1,1;
  1;
  2,1;
  ...
The multiset of positive exponents in the prime factorization of 6 = 2*3 is {1,1} (1s are often left implicit as exponents). The prime signature of 6 is therefore {1,1}.
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, as does 18 = 2*3^2. Rows 12 and 18 of the table both read {2,1}.

Jason Kimberley, Table of i, a(i) for i = 2..24301 (n = 2..10000)

Cf. A025487, A001221 (row lengths), A001222 (row sums). A118914 gives the nondecreasing version. A124010 lists exponents in n's prime factorization in natural order, with A124010(1) = 0.
A212172 cross-references over 20 sequences that depend solely on n's prime exponents >= 2, including the "core" sequence A000688.  Other sequences determined by the exponents in the prime factorization of n include:
Multiplicative: A000005, A007425, A008683, A008836, A034444, A037445, A181819.
Additive: A001221, A001222, A056169.
Other: A001055, A008480, A010553, A038548, A050320, A051707, A071625, A074206, A076078, A085082, A088873.
A highly incomplete selection of sequences, each definable by the set of prime signatures possessed by its members: A000040, A000290, A000578, A000583, A000961, A001248, A001358, A001597, A001694, A002808, A004709, A005117, A006881, A013929, A030059, A030229, A052486.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)])):n in[1..76]]; // Jason Kimberley, Jun 13 2012

apply( {A212171_row(n)=vecsort(factor(n)[,2]~,,4)}, [1..40])\\ M. F. Hasler, Apr 19 2022

Row n of A118914, reversed.

Row n of A124010 for n > 1, with exponents sorted in nonincreasing order. Equivalently, row A046523(n) of A124010 for n > 1.

A036450 a(n) = d(d(d(n))), the 3rd iterate of the number-of-divisors function with an initial value of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 2, 4, 2, 3, 3, 2, 2, 3, 2, 3, 3

Offset: 1

Labos Elemer

nonn

The iterated d function rapidly converges to the fixed point 2.
From N. J. A. Sloane, Jun 02 2014: (Start)
The fourth iterate begins as follows:
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 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, 3, 2, 2, 2, 2, 2, 3, 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, ... . (End)

			n = 5040, d(5040) = 60, d(d(5040)) = d(60) = 12 and a(5040) = d(12) = 6.

S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 128. - N. J. A. Sloane, Jun 02 2014

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000
R. Bellman and H. N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340.

Cf. A000005, A010553, A036452, A036453.

f[n_]:=Length[Divisors[n]];Table[Nest[f,n,3],{n,6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)

a(n)=numdiv(numdiv(numdiv(n))) \\ Charles R Greathouse IV, Nov 16 2022

from sympy import divisor_count
def A036450(n): return divisor_count(divisor_count(divisor_count(n))) # Chai Wah Wu, Nov 17 2022

A036452 a(n) = d(d(d(d(n)))), the 4th iterate of number-of-divisors function with initial value of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer

nonn

The iterated d function rapidly converges to fixed point 2. For k=4, the first n for which a(n)>2 is 60.

			E.g., n=96 and its successive iterates are 12,6,4,3 and 2. The 4th term is a(96)=3.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000

Cf. A000005, A010553, A036450, A036453.

f[n_]:=DivisorSigma[0,n]; Table[Nest[f,n,4], {n,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)

a(n)=my(d=numdiv);d(d(d(d(n)))) \\ Charles R Greathouse IV, Apr 07 2012

a(n) = d(d(d(d(n)))).

A036453 a(n) = d(d(d(d(d(n))))), the 5th iterate of the number-of-divisors function d = A000005, with initial value n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer

nonn

The iterated d function rapidly converges to fixed point 2. In the 5th iterated d-sequence, the first term different from the fixed point 2 appears at n = 5040. The 6th and further iterated sequences have very long initial segment of 2's. In the 6th one the first non-stationary term is a(293318625600) = 3. In such sequences any large value occurs infinite many times and constructible.

Differs from A007395 for n = 1, 5040, 7920, 8400, 9360, 10080, 10800, etc. - R. J. Mathar, Oct 20 2008

			E.g., n = 96 and its successive iterates are 12, 6, 4, 3 and 2. The 5th term is a(96) = 2 is stationary (fixed).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A010553, A036450, A036452.

Table[Nest[DivisorSigma[0,#]&,n,5],{n,110}] (* Harvey P. Dale, Jun 18 2021 *)

a(n)=my(d=numdiv);d(d(d(d(d(n))))) \\ Charles R Greathouse IV, Apr 07 2012

Previous Mathematica program replaced by Harvey P. Dale, Jun 18 2021

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

A141113 Positive integers k such that d(d(k)) divides k, where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 2, 4, 6, 12, 15, 16, 20, 21, 24, 27, 28, 32, 33, 36, 39, 40, 44, 48, 51, 52, 56, 57, 60, 64, 68, 69, 72, 76, 80, 84, 87, 88, 90, 92, 93, 96, 104, 108, 111, 112, 116, 120, 123, 124, 126, 128, 129, 132, 136, 141, 144, 148, 150, 152, 156, 159, 164, 172, 176, 177, 180

Offset: 1

Leroy Quet, Jun 04 2008

nonn

			28 has 6 divisors and 6 has 4 divisors. 4 divides 28, so 28 is in the sequence.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A010553, A141114, A141115.

with(numtheory): a:=proc(n) if `mod`(n, tau(tau(n))) = 0 then n else end if end proc: seq(a(n),n=1..200); # Emeric Deutsch, Jun 05 2008

Select[Range[200],Divisible[#,DivisorSigma[0,DivisorSigma[0,#]]]&] (* Harvey P. Dale, Feb 05 2012 *)

is(k) = k%numdiv(numdiv(k)) == 0; \\ Jinyuan Wang, Feb 19 2019

More terms from Emeric Deutsch, Jun 05 2008

A141114 Positive integers k where d(d(k)) is coprime to k, where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 45, 46, 47, 49, 53, 55, 58, 59, 61, 62, 63, 65, 67, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 89, 91, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 113, 115, 117, 118, 119, 121

Offset: 1

Leroy Quet, Jun 04 2008

nonn

Includes all primes, squares of odd primes, and squarefree semiprimes coprime to 3. - Robert Israel, Dec 16 2019

			26 has 4 divisors and 4 has 3 divisors. 3 is coprime to 26, so 26 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A010553, A141113, A141115.

[k:k in [1..130]|Gcd(k,#Divisors(#Divisors(k))) eq 1]; // Marius A. Burtea, Dec 16 2019

filter:= proc(n) uses numtheory;
  igcd(tau(tau(n)), n) = 1
end proc:
select(filter, [$1..200]); # Robert Israel, Dec 16 2019

Select[Range[200], GCD[DivisorSigma[0, DivisorSigma[0, # ]], # ] == 1 &] (* Stefan Steinerberger, Jun 05 2008 *)

is(n) = gcd(numdiv(numdiv(n)), n)==1 \\ Felix Fröhlich, Dec 16 2019

More terms from Stefan Steinerberger, Jun 05 2008

A141115 Those positive integers k where both d(d(k)) is not coprime to k and d(d(k)) does not divide k, where d(k) is the number of divisors of k.

Original entry on oeis.org

18, 30, 42, 50, 54, 66, 70, 78, 98, 102, 110, 114, 130, 138, 140, 154, 160, 162, 170, 174, 182, 186, 190, 200, 220, 222, 224, 230, 238, 242, 246, 250, 258, 260, 266, 282, 286, 290, 308, 310, 315, 318, 322, 338, 340, 350, 352, 354, 364, 366, 370, 374, 380, 392

Offset: 1

Leroy Quet, Jun 04 2008

nonn

			50 has 6 divisors and 6 has 4 divisors. 4 is not coprime to 50 and 4 does not divide 50. So 50 is in the sequence.

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

Cf. A010553, A141113, A141114.

Select[Range[400], GCD[DivisorSigma[0,DivisorSigma[0, # ]], # ] > 1 && Mod[ #, DivisorSigma[0, DivisorSigma[0, # ]]] > 0 &] (* Stefan Steinerberger, Jun 05 2008 *)

More terms from Stefan Steinerberger, Jun 05 2008

cp's OEIS Frontend

A335831 Numbers k with a record value of tau(tau(k)) (A010553), where tau(k) is the number of divisors of k (A000005).

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A083399 Number of divisors of n that are not divisors of other divisors of n.

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A212171 Prime signature of n (nonincreasing version): row n of table lists positive exponents in canonical prime factorization of n, in nonincreasing order.

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A036450 a(n) = d(d(d(n))), the 3rd iterate of the number-of-divisors function with an initial value of n.

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A036452 a(n) = d(d(d(d(n)))), the 4th iterate of number-of-divisors function with initial value of n.

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A036453 a(n) = d(d(d(d(d(n))))), the 5th iterate of the number-of-divisors function d = A000005, with initial value 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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