A054896 a(n) = Sum_{k>0} floor(n/7^k).

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

Offset: 0

Author

Henry Bottomley, May 23 2000

Keywords

• nonn

Comments

Exponent of the highest power of 7 dividing n!.

Examples

			  a(10^0) = 0.
  a(10^1) = 1.
  a(10^3) = 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
		

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. A053828, A054893, A054895, A054899, A067080, A098844, A132031, A214411.

Cf. A000142, A214411.

Programs

Magma

function A054896(n)
  if n eq 0 then return n;
  else return A054896(Floor(n/7)) + Floor(n/7);
  end if; return A054896;
end function;
[A054896(n): n in [0..100]]; // G. C. Greubel, Feb 09 2023

Mathematica

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

PARI

a(n)=(n-sumdigits(n, 7))\6 \\ Alan Michael Gómez Calderón, Oct 08 2024

SageMath

def A054896(n):
    if (n==0): return 0
    else: return A054896(n//7) + (n//7)
[A054896(n) for n in range(101)] # G. C. Greubel, Feb 09 2023

Formula

a(n) = 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)

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

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

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

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.: (1/(1-x))*Sum_{k > 0} x^(7^k)/(1-x^(7^k)). (End)

Partial sums of A214411. - R. J. Mathar, Jul 08 2021

a(n) = A214411(A000142(n)). - Michel Marcus, Oct 07 2024

Extensions

Examples added by Hieronymus Fischer, Jun 06 2012