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
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