A122841 Greatest k such that 6^k divides n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Sep 13 2006

Keywords

• nonn

Comments

See A054895 for the partial sums. - Hieronymus Fischer, Jun 08 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A122840, A007814, A007949, A054895, A234959.

Programs

Haskell

a122841 = f 0 where
   f y x = if r > 0 then y else f (y + 1) x'
           where (x', r) = divMod x 6
-- Reinhard Zumkeller, Nov 10 2013

Mathematica

Table[IntegerExponent[n, 6], {n, 1, 100}] (* Amiram Eldar, Sep 14 2020 *)

PARI

a(n) = valuation(n, 6); \\ Michel Marcus, Jan 17 2022

Formula

From Hieronymus Fischer, Jun 03 2012: (Start)

With m = floor(log_6(n)), frac(x) = x-floor(x):

a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/6^j))).

a(n) = m + Sum_{j=1..m} (floor(-frac(n/6^j))).

a(n) = A054895(n) - A054895(n-1).

G.f.: Sum_{j>0} x^6^j/(1-x^6^j). (End)

a(A047253(n)) = 0; a(A008588(n)) > 0; a(A044102(n)) > 1. - Reinhard Zumkeller, Nov 10 2013

6^a(n) = A234959(n), n >= 1. - Wolfdieter Lang, Jun 30 2014

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/5. - Amiram Eldar, Jan 17 2022

a(n) = min(A007814(n), A007949(n)). - Jianing Song, Jul 23 2022