A132031 -id:A132031 - cp's OEIS frontend

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

Henry Bottomley, May 23 2000

nonn

Exponent of the highest power of 7 dividing n!.

			  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

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

Cf. A011371 and A054861 for analogs involving powers of 2 and 3.
Cf. A053828, A054893, A054895, A054899, A067080, A098844, A132031, A214411.
Cf. A000142, A214411.

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

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

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

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

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

Examples added by Hieronymus Fischer, Jun 06 2012

A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

Original entry on oeis.org

4, 5, 8, 7, 6, 6, 7, 2, 6, 6, 9, 9, 7, 6, 8, 9, 8, 5, 0, 2, 0, 0, 0, 5, 1, 5, 3, 3, 6, 9, 7, 4, 3, 7, 2, 1, 7, 8, 2, 5, 4, 6, 6, 8, 8, 7, 1, 4, 7, 3, 1, 8, 7, 0, 0, 7, 8, 2, 4, 4, 0, 1, 3, 8, 5, 0, 6, 9, 9, 7, 4, 4, 0, 3, 2, 6, 5, 9, 3, 0, 3, 6, 5, 2, 3, 7, 8, 1, 7, 1, 0, 9, 0, 4, 0, 5, 8, 4, 7, 5, 9, 8, 2

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.4587667266997689850200...

Cf. A048651, A098844, A067080, A132019, A132026, A132031, A132035, A000217, A109808.

digits = 103; NProduct[1-1/(2*7^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/7], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_7(n))} floor(n/7^k)*7^k/n.
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^(1/2*(1+floor(log_7(n)))*floor(log_7(n))).
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^A000217(floor(log_7(n))).
Equals 1/2*exp(-Sum_{n>0} 7^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132031(n)/A132031(n+1).
Equals Product_{n>=1} (1 - 1/A109808(n)). - Amiram Eldar, May 08 2023

cp's OEIS Frontend

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

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