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A054895
a(n) = Sum_{k>0} floor(n/6^k).
11
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
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.
Sequence in context: A097992 A195177 A147583 * A194699 A262694 A137588
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