A386262 -id:A386262 - 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).

cp's OEIS Frontend

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

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