A010553 -id:A010553 - cp's OEIS frontend

A386261 a(n) = A001511(A001511(n)), where A001511 is the ruler function.

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

Offset: 1

Amiram Eldar, Jul 17 2025

The first occurrence of k = 1, 2, ... is at n = 2^(2^(k-1) - 1) = A058891(k).

The asymptotic density of the occurrences of k = 1, 2, ... in this sequence is 2^(2^(k-1))/(2^(2^k)-1) = 2/3, 4/15, 16/255, 256/65535, 65536/4294967295, ...

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

Cf. A001511, A003159, A048649, A058891.

Similar sequences: A010553, A010554, A051027, A055033, A058699, A068346, A132090, A178601, A181776, A284908, A386262.

f[n_] := IntegerExponent[n, 2] + 1; a[n_] := f[f[n]]; Array[a, 100]

a(n) = valuation(valuation(n, 2) + 1, 2) + 1;

a(n) >= 1, with equality if and only if n is in A003159.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{m>=0} 1/(2^(2^m) - 1) = 1.4039368... (A048649).

A386262 a(n) = A051903(A051903(n)) for n >= 2, a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 17 2025

The first occurrence of k = 1, 2, ... is at n = 2^(2^k) = A001146(k).
If n is an exponentially squarefree number (A209061) then a(n) <= 1. The converse is not necessarily true, with n = 2592 = 2^5 * 3^4 being the least counterexample.
The asymptotic density of the occurrences of 0 in this sequence is 1/zeta(2) = 6/Pi^2 (A059956).
The asymptotic density of the occurrences of 1 in this sequence is Sum_{k squarefree > 1} (1/zeta(k+1) - 1/zeta(k)) = 0.348423339572619656701... .

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

Cf. A005117, A001146, A051903, A059956, A209061.

Similar sequences: A010553, A010554, A051027, A055033, A058699, A068346, A132090, A178601, A181776, A284908, A386261.

f[n_] := Max[FactorInteger[n][[;; , 2]]]; f[1] = 0; a[n_] := f[f[n]]; a[1] = 0; Array[a, 100]

f(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = if(n == 1, 0, f(f(n)));

a(n) = 0 if and only if n is squarefree (A005117).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} A051903(k) * (1/zeta(k+1)-1/zeta(k)) = 0.43779421197744649258... .

A053023 Divisor function applied thrice to n!.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 6, 6, 6, 5, 8, 8, 8, 8, 8, 9, 8, 10, 8, 10, 10, 12, 10, 10, 10, 15, 16, 16, 18, 16, 8, 16, 10, 12, 16, 18, 14, 3, 14, 16, 24, 16, 16, 20, 24, 24, 28, 24, 24, 16, 24, 24, 32, 32, 28, 32, 18, 20, 24, 28, 36, 32, 36, 24, 21, 24, 20, 40, 40, 30, 30, 36, 40, 42, 24, 32

Offset: 1

Labos Elemer, Feb 24 2000

nonn

			A036450(x) = 8 if x = 11!, 12!, 13!, 14!, 15!. The reversed sequences of these d-iterates are {2, 3, 4, 8, 24, 540, 39916800}, {2, 3, 4, 8, 24, 792, 479001600}, {2, 3, 4, 8, 30, 1584, 6227020800}, {2, 3, 4, 8, 30, 2592, 87178291200}, {2, 3, 4, 8, 42, 4032, 1307674368000}.

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

Cf. A000005, A000142, A010553, A036450, A053021.

a[n_] := Nest[DivisorSigma[0, #]&, n!, 3]; Array[a, 100] (* Amiram Eldar, Aug 12 2024 *)

a(n) = numdiv(numdiv(numdiv(n!))); \\ Amiram Eldar, Aug 12 2024

a(n) = d(d(d(n!))) = A036450(A000142(n)) = A036450(n!).

A140128 A positive integer k is included if d(d(k)) = d(d(k+1)), where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 3, 4, 14, 16, 21, 26, 33, 34, 35, 38, 44, 57, 75, 85, 86, 93, 94, 98, 104, 116, 118, 122, 133, 135, 141, 142, 145, 147, 152, 153, 158, 164, 170, 171, 174, 175, 177, 188, 189, 201, 202, 205, 207, 213, 214, 217, 218, 225, 230, 231, 242, 243, 244, 245, 253, 272

Offset: 1

Leroy Quet, Jun 04 2008

nonn

			35 has 4 divisors and 4 has 3 divisors. 36 has 9 divisors and 9 has 3 divisors. Since d(d(35)) = d(d(36)) (=3), then 35 is included in the sequence.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005, A010553.

A005237 is a subsequence.

Select[Range[250], DivisorSigma[0, DivisorSigma[0, # ]] == DivisorSigma[0, DivisorSigma[0, # + 1]] &] (* Stefan Steinerberger, Jun 05 2008 *)

is(k) = numdiv(numdiv(k)) == numdiv(numdiv(k+1)); \\ Amiram Eldar, Apr 16 2024

For m = 2,3,4,5..., a(m) is the smallest integer > a(m-1) such that A010553(a(m)) = A010553(a(m)+1).

More terms from Stefan Steinerberger, Jun 05 2008

a(56)-a(57) from Ray Chandler, Jun 26 2009

A163374 a(n) = tau(tau(phi(n))).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 3, 3, 3, 2, 3, 4, 3, 4, 3, 3, 3, 4, 4, 4, 4, 4, 3, 4, 2, 4, 2, 4, 4, 3, 4, 4, 2, 4, 4, 4, 4, 4, 3, 3, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3, 2, 6, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 3, 4, 3, 6, 4, 4, 4, 4, 4, 3, 4, 2, 4

Offset: 1

Jaroslav Krizek, Jul 25 2009

			tau(tau(phi(7))) = tau(tau(6)) = tau(4) = 3. Thus a(7) = 3. - _Derek Orr_, Jul 27 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005 (tau), A000010 (phi), A010553, A062821.

[NumberOfDivisors(NumberOfDivisors(EulerPhi(n))): n in [1..100]]; // Vincenzo Librandi, Jul 27 2014

DivisorSigma[0,DivisorSigma[0,EulerPhi[Range[90]]]] (* Harvey P. Dale, Mar 25 2016 *)

a(n)=sigma(sigma(eulerphi(n),0),0); \\ Derek Orr, Jul 27 2014

a(n) = A000005(A000005(A000010(n))) = A000005(A062821(n)) = A010553(A000010(n)).

More terms from Vincenzo Librandi, Jul 27 2014

A173338 Numbers n such that tau(tau(n)) = sopf(sopf(n)), sopf = A008472.

Original entry on oeis.org

2, 4, 14, 16, 27, 64, 158, 168, 196, 216, 312, 378, 384, 440, 456, 482, 546, 680, 702, 744, 770, 1024, 1026, 1032, 1160, 1454, 1608, 1640, 1674, 1880, 2024, 2058, 2295, 2322, 2472, 2750, 2805, 2944, 3336, 3560, 3608, 3618, 3768, 3828, 3944, 3960, 4040, 4096

Offset: 1

Michel Lagneau, Feb 16 2010

nonn

sopf(n) is the sum of distinct primes dividing n (without repetition, A008472), tau(n) is the number of divisors of n (A000005).

			4 is in the sequence: tau(4) = 3, tau(3) = 2; sopf(4) = 2, sopf(2) = 2.
546 is in the sequence: tau(546) = 16, tau(16) = 5; sopf(546) = 25, sopf(25) = 5.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
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].
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.

Cf. A000005, A008472, A010553.

f:=func; g:=func; [k:k in [2..5000]|f(f(k)) eq g(g(k)) ]; // Marius A. Burtea, Nov 14 2019

with(numtheory): for n from 1 to 60000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)): tt1:= ifactors(t2)[2] : tt2 :=sum(tt1[i][1], i=1..nops(tt1)):if tau(tau(n))= tt2 then print (n): else fi : od :
# second Maple program:
with(numtheory): sopf:= n-> add(i, i=factorset(n)):
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 0, a(n-1))
      while tau(tau(k)) <> sopf(sopf(k)) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 26 2010

Select[Range[4100],DivisorSigma[0,DivisorSigma[0,#]]==Total[ Transpose[ FactorInteger[ Total[Transpose[FactorInteger[#]][[1]]]]][[1]]]&] (* Harvey P. Dale, Aug 05 2013 *)

{ n : A010553(n) = A008472(A008472(n)) }.

Corrected and edited by Michel Lagneau, Apr 25 2010

A173746 Numbers k such that tau(tau(k)) = rad(k).

Original entry on oeis.org

1, 2, 4, 16, 27, 64, 72, 96, 108, 288, 432, 486, 648, 768, 972, 1024, 1536, 1728, 3456, 4096, 5832, 6561, 13122, 17496, 20736, 24576, 27648, 39366, 41472, 65536, 98304, 104976, 110592, 147456, 186624, 256000, 262144, 314928, 400000, 419904, 472392

Offset: 1

Michel Lagneau, Feb 23 2010

nonn

Tau = A000005 is the number of divisors of its argument. rad(n) = A007947(n) is the product of the primes dividing n.

Note that rad() is idempotent: rad(rad(n)) = rad(n). - R. J. Mathar, Nov 07 2011

			288 is in the sequence because tau(288)= 18, tau(18)=6, rad(288)=6.

Donovan Johnson, Table of n, a(n) for n = 1..496 (terms < 10^18)

A010553 := proc(n)
        numtheory[tau](numtheory[tau](n)) ;
end proc:
for n from 1 to 480000 do
        if A010553(n) = A007947(n) then
                printf("%d,",n) ;
        end if;
end do: # R. J. Mathar, Nov 07 2011

{n : A010553(n) = A007947(n)}.

Example corrected and edited by Michel Lagneau, Apr 25 2010

A280583 a(n) = product of divisors of the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 8, 2, 8, 3, 8, 2, 36, 2, 8, 8, 5, 2, 36, 2, 36, 8, 8, 2, 64, 3, 8, 8, 36, 2, 64, 2, 36, 8, 8, 8, 27, 2, 8, 8, 64, 2, 64, 2, 36, 36, 8, 2, 100, 3, 36, 8, 36, 2, 64, 8, 64, 8, 8, 2, 1728, 2, 8, 36, 7, 8, 64, 2, 36, 8, 64, 2, 1728, 2, 8, 36, 36, 8

Offset: 1

Jaroslav Krizek, Jan 07 2017

nonn

			For n = 6; a(n) = product of divisors (tau(6)) = 1*2*4 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A001248, A007955, A010553, A062069.

[&*[d: d in Divisors(#[d: d in Divisors(n)])]: n in [1..100]]

Table[Times@@Divisors[DivisorSigma[0,n]],{n,80}] (* Harvey P. Dale, Dec 04 2021 *)

from math import isqrt
from sympy import divisor_count
def A280583(n): return (lambda m:(isqrt(m) if (c:=divisor_count(m)) & 1 else 1)*m**(c//2))(divisor_count(n)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(A000005(n)).
a(p) = 2 for p = primes (A000040).
a(n) = 3 for squares of primes (A001248).

A293167 a(n) = Sum_{k = 1..n} d(d(d(k))), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 24, 26, 28, 30, 32, 34, 37, 39, 42, 44, 46, 48, 51, 53, 55, 57, 60, 62, 65, 67, 70, 72, 74, 76, 78, 80, 82, 84, 87, 89, 92, 94, 97, 100, 102, 104, 107, 109, 112, 114, 117, 119, 122, 124, 127, 129, 131, 133, 137, 139, 141, 144, 146, 148, 151, 153, 156, 158

Offset: 1

N. J. A. Sloane, Oct 17 2017

nonn

David A. Corneth, Table of n, a(n) for n = 1..10000
Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340.
Imre Kátai, On the iteration of the divisor-function, Publ. Math. Debrecen, Vol. 16 (1969), pp. 3-15.

Part of the sequence A000005, A006218, A010553, A036450, A139130.

Accumulate[Table[DivisorSigma[0, DivisorSigma[0, DivisorSigma[0, n]]], {n, 80}]] (* Alonso del Arte, Oct 17 2017 *)

a(n) = sum(k=1, n, numdiv(numdiv(numdiv(k)))); \\ Michel Marcus, Oct 17 2017

first(n) = {my(v = vector(n)); v[1] = 1; for(i=2,n,v[i] = v[i-1] + numdiv(numdiv(numdiv(i)))); v} \\ David A. Corneth, Oct 17 2017

a(1) = 1; a(n + 1) = a(n) + A036450(n + 1) for n > 0. - David A. Corneth, Oct 17 2017

a(n) = (1 + o(1)) * c * n * log(log(log(n))), where c > 0 is a constant (Kátai, 1969). - Amiram Eldar, Apr 17 2024

A343502 Numbers k such that tau(tau(k)) and tau(k+1) are both prime, where tau is the number of divisors function.

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 15, 16, 22, 36, 46, 58, 82, 100, 106, 120, 166, 168, 178, 196, 210, 226, 256, 262, 270, 280, 312, 330, 346, 358, 378, 382, 408, 456, 462, 466, 478, 502, 520, 546, 562, 570, 586, 616, 640, 676, 690, 718, 728, 750, 760, 838, 858, 862, 886

Offset: 1

Claude H. R. Dequatre, Apr 17 2021

Considering the first 10^8 positive integers there are 1439855 terms in the sequence and only the first two (2,3) are prime, all the others are composite numbers of which only three are odd (15, 65535 and 4194303).
Conjecture: all members except 2 and 3 are composite.
Open question: is there a finite number of odd terms in this sequence?

			16 is a term because tau(16) = 5 and tau(5) = 2 and tau(17) = 2 and 2 is prime.
23 is not a term because tau(23) = 2 and tau(2) = 2 and tau(24) = 8 and 2 is prime but not 8.
98 is not a term because tau(98) = 6 and tau(6) = 4 and tau(99) = 6 and 4 and 6 are not prime.

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

Cf. A000005, A000040, A010553. Includes A077065.

filter:= proc(n)
  isprime(numtheory:-tau(n+1)) and isprime(numtheory:-tau(numtheory:-tau(n)))
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2025

With[{t = DivisorSigma}, Select[Range[1000], And @@ PrimeQ[{t[0, t[0, #]], t[0, # + 1]}] &]] (* Amiram Eldar, May 27 2021 *)

for(k=1,1e4,if(isprime(numdiv(numdiv(k))) && isprime(numdiv(k+1)),print1(k", ")))

cp's OEIS Frontend

A386261 a(n) = A001511(A001511(n)), where A001511 is the ruler function.

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A386262 a(n) = A051903(A051903(n)) for n >= 2, a(1) = 0.

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A053023 Divisor function applied thrice to n!.

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A140128 A positive integer k is included if d(d(k)) = d(d(k+1)), where d(k) is the number of divisors of k.

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A163374 a(n) = tau(tau(phi(n))).

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A173338 Numbers n such that tau(tau(n)) = sopf(sopf(n)), sopf = A008472.

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A173746 Numbers k such that tau(tau(k)) = rad(k).

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