A054895 a(n) = Sum_{k>0} floor(n/6^k).

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16

Offset

0,13

Comments

Different from the highest power of 6 dividing n! (cf. A054861). - Hieronymus Fischer, Aug 14 2007

Partial sums of A122841. - Hieronymus Fischer, Jun 06 2012

Links

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor(n/6) + floor(n/36) + floor(n/216) + floor(n/1296) + ...

a(n) = (n - A053827(n))/5.

From Hieronymus Fischer, Aug 14 2007: (Start)

a(n) = a(floor(n/6)) + floor(n/6).

a(6*n) = n + a(n).

a(n*6^m) = n*(6^m-1)/5 + a(n).

a(k*6^m) = k*(6^m-1)/5, for 0 <= k < 6, m >= 0.

Asymptotic behavior:

a(n) = (n/5) + O(log(n)).

a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.

a(n) <= (n-1)/5; equality holds for powers of 6.

a(n) >= ((n-5)/5) - floor(log_6(n)); equality holds for n=6^m-1, m>0.

lim inf (n/5 - a(n)) = 1/5, for n-->oo.

lim sup (n/5 - log_6(n) - a(n)) = 0, for n-->oo.

lim sup (a(n+1) - a(n) - log_6(n)) = 0, for n-->oo.

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

Example

  a(10^0) = 0.
  a(10^1) = 1.
  a(10^2) = 18.
  a(10^3) = 197.
  a(10^4) = 1997.
  a(10^5) = 19996.
  a(10^6) = 199995.
  a(10^7) = 1999995.
  a(10^8) = 19999994.
  a(10^9) = 199999993.

Mathematica

Table[t=0; p=6; While[s=Floor[n/p]; t=t+s; s>0, p *= 6]; t, {n, 0, 100}]

Prog

(Haskell)
a054895 n = a054895_list !! n
a054895_list = scanl (+) 0 a122841_list
-- Reinhard Zumkeller, Nov 10 2013
(Magma)
function A054895(n)
  if n eq 0 then return n;
  else return A054895(Floor(n/6)) + Floor(n/6);
  end if; return A054895;
end function;
[A054895(n): n in [0..100]]; // G. C. Greubel, Feb 09 2023
(SageMath)
def A054895(n):
    if (n==0): return 0
    else: return A054895(n//6) + (n//6)
[A054895(n) for n in range(104)] # G. C. Greubel, Feb 09 2023

Crossrefs

Cf. A011371 and A054861 for analogs involving powers of 2 and 3.

Cf. A053827, A054861, A054899, A067080, A098844, A122841, A132030.

Sequence in context: A097992 A195177 A147583 * A194699 A262694 A137588

Adjacent sequences: A054892 A054893 A054894 * A054896 A054897 A054898

Keyword

nonn

Author

Henry Bottomley, May 23 2000

Extensions

An incorrect formula was deleted by N. J. A. Sloane, Nov 18 2008

Examples added by Hieronymus Fischer, Jun 06 2012

Status

approved