A110591 Number of digits in base-4 representation of n.

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

Offset: 0

Author

Jonathan Vos Post, Jul 29 2005

Keywords

• easy
• nonn

Comments

Number of digits in A007090(n).

In terms of the repetition convolution operator #, where (sequence A) # (sequence B) = the sequence consisting of A(n) copies of B(n), this sequence is the repetition convolution A110594 # n. Over the set of positive infinite integer sequences, # gives a nonassociative noncommutative groupoid (magma) with a left identity (A000012) but no right identity, where the left identity is also a right nullifier and idempotent. For any positive integer constant c, the sequence c*A000012 = (c,c,c,c,...) is also a right nullifier; for c = 1, this is A000012; for c = 3 this is A010701.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000012, A007090, A010701, A049804, A081604, A110594.

Programs

Haskell

import Data.List (unfoldr)
a110591 0 = 1
a110591 n = length $
   unfoldr (\x -> if x == 0 then Nothing else Just (x, x `div` 4)) n
-- Reinhard Zumkeller, Apr 22 2011

Maple

A110592 := proc(n)
    if n = 0 then
        1;
    else
        1+floor(log[4](n)) ;
    end if;
end proc:
seq(A110592(n),n=0..50) ; # R. J. Mathar, Sep 02 2020

Mathematica

a[n_] := If[n == 0, 1, Floor[Log[4, n]] + 1];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 24 2020 *)

Formula

G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(4^k). - Ilya Gutkovskiy, Jan 08 2017

a(n) = floor(log_4(n)) + 1 for n >= 1. - Petros Hadjicostas, Dec 12 2019