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A054896 a(n) = Sum_{k>0} floor(n/7^k). 12
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,15

COMMENTS

Highest power of 7 dividing n!.

LINKS

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

FORMULA

floor[n/7] + floor[n/49] + floor[n/343] + floor[n/2401] + ....

a(n) = (n - A053828(n))/6.

From Hieronymus Fischer, Aug 14 2007: (Start)

Recurrence:

a(n) = floor(n/7) + a(floor(n/7));

a(7*n) = n + a(n);

a(n*7^m) = n*(7^m-1)/6+a(n).

a(k*7^m) = k*(7^m-1)/6, for 0<=k<7, m>=0.

Asymptotic behavior:

a(n) = n/6 + O(log(n)),

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

a(n) <= (n-1)/6; equality holds for powers of 7.

a(n) >= (n-6)/6 - floor(log_7(n)); equality holds for n=7^m-1, m>0. -

lim inf (n/6 - a(n)) = 1/6, for n-->oo.

lim sup (n/6 - log_7(n) - a(n)) = 0, for n-->oo.

lim sup (a(n+1) - a(n) - log_7(n)) = 0, for n-->oo.

G.f.: g(x) = sum{k>0, x^(7^k)/(1-x^(7^k))}/(1-x). (End)

EXAMPLE

a(100)=16.

a(10^3)=164.

a(10^4)=1665.

a(10^5)=16662.

a(10^6)=166664.

a(10^7)=1666661.

a(10^8)=16666662.

a(10^9)=166666661

MATHEMATICA

Table[t = 0; p = 7; While[s = Floor[n/p]; t = t + s; s > 0, p *= 7]; t, {n, 0, 100} ]

CROSSREFS

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

Cf. A054895, A054899, A067080, A098844, A132031.

Sequence in context: A132270 A195174 A187185 * A052364 A052374 A003074

Adjacent sequences:  A054893 A054894 A054895 * A054897 A054898 A054899

KEYWORD

nonn

AUTHOR

Henry Bottomley, May 23 2000

EXTENSIONS

Examples added by Hieronymus Fischer, Jun 06 2012

STATUS

approved

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Last modified July 21 21:14 EDT 2019. Contains 325199 sequences. (Running on oeis4.)