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

Author

Henry Bottomley, May 23 2000

Keywords

• nonn

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

Examples

			  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.
		

Links

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

Crossrefs

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

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

Programs

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

Mathematica

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

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

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)

Extensions

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

Examples added by Hieronymus Fischer, Jun 06 2012