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

1,2

Comments

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, ...

Links

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

Formula

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).

Multiplicative with a(2^e) = A001511(e+1), and a(p^e) = 1 for an odd prime p. - Amiram Eldar, Oct 31 2025

Mathematica

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

Prog

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

Crossrefs

Cf. A001511, A003159, A048649, A058891.

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

Sequence in context: A393663 A185147 A206921 * A327402 A123529 A140747

Adjacent sequences: A386258 A386259 A386260 * A386262 A386263 A386264

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jul 17 2025

Status

approved