A181776 -id:A181776 - 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... .

A366779 a(n) = lambda(lambda(lambda(n))), where lambda(n) is the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 4, 10, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 6, 1, 2, 2, 1, 2, 1, 2, 4, 2, 4, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 6, 2, 4, 1, 2, 2, 2

Offset: 1

Miles Englezou, Dec 15 2023

nonn

			a(5) = 1, since A181776(5) = 2, and A002322(2) = 1.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
N. Harland, The iterated Carmichael lambda function, arXiv:1111.3667 [math.NT], 2011.
G. Martin and C. Pomerance, The iterated Carmichael lambda-function and the number of cycles of the power generator, Acta Arith. 118:4 (2005), pp. 305-335.

Cf. A002322 (lambda function), A181776 (lambda function at two iterations).

a:= n-> (numtheory[lambda]@@3)(n):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 19 2024

a[n_]:=Nest[CarmichaelLambda,n,3]; Array[a,87] (* Stefano Spezia, Jan 20 2024 *)

a(n) = lcm(znstar(lcm(znstar(lcm(znstar(11)[2]))[2]))[2])

from sympy import reduced_totient
def A366779(n): return reduced_totient(reduced_totient(reduced_totient(n))) # Chai Wah Wu, Jan 29 2024

a(n) = A002322(A181776(n)).

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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A366779 a(n) = lambda(lambda(lambda(n))), where lambda(n) is the Carmichael lambda function (A002322).

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