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

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

Author

Amiram Eldar, Jul 17 2025

Keywords

• nonn
• easy

Comments

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... .

Links

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

Crossrefs

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

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

Programs

Mathematica

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

PARI

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

Formula

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... .