A004128 a(n) = Sum_{k=1..n} floor(3*n/3^k).

0, 1, 2, 4, 5, 6, 8, 9, 10, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 30, 31, 32, 34, 35, 36, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 55, 57, 58, 59, 61, 62, 63, 66, 67, 68, 70, 71, 72, 74, 75, 76, 80, 81, 82, 84, 85, 86, 88, 89, 90, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

3-adic valuation of (3n)!; cf. A054861.

Denominators of expansion of (1-x)^{-1/3} are 3^a(n). Numerators are in |A067622|.

References

• Gary W. Adamson, in "Beyond Measure, A Guided Tour Through Nature, Myth and Number", by Jay Kappraff, World Scientific, 2002, p. 356.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A004117, A001511, A051064, A055457.

Cf. A054861, A067080, A098844, A132027, A005187, A054899.

Programs

Haskell

a004128 n = a004128_list !! (n-1)
a004128_list = scanl (+) 0 a051064_list
-- Reinhard Zumkeller, May 23 2013

Magma

[n + Valuation(Factorial(n), 3): n in [0..70]]; // Vincenzo Librandi, Jun 12 2019

Maple

A004128 := proc(n)
    A054861(3*n) ;
end proc:
seq(A004128(n),n=0..100) ; # R. J. Mathar, Nov 04 2017

Mathematica

Table[Total[NestWhileList[Floor[#/3] &, n, # > 0 &]], {n, 0, 70}] (* Birkas Gyorgy, May 20 2012 *)
A004128 = Log[3, CoefficientList[ Series[1/(1+x)^(1/3), {x, 0, 100}], x] // Denominator] (* Jean-François Alcover, Feb 19 2015 *)
Flatten[{0, Accumulate[Table[IntegerExponent[3*n, 3], {n, 1, 100}]]}] (* Vaclav Kotesovec, Oct 17 2019 *)

PARI

{a(n) = my(s, t=1); while(t<=n, s += n\t; t*=3);s}; /* Michael Somos, Feb 26 2004 */

PARI

a(n) = (3*n-sumdigits(n,3))/2; \\ Christian Krause, Jun 10 2025

Python

def A007949(n):
    c = 0
    while not (a:=divmod(n,3))[1]:
        c += 1
        n = a[0]
    return c
def A004128(n): return n+sum(A007949(i) for i in range(3,n+1)) # Chai Wah Wu, Feb 28 2025

Sage

A004128 = lambda n: A004128(n//3) + n if n > 0 else 0
[A004128(n) for n in (0..70)]  # Peter Luschny, Nov 16 2012

Formula

A051064(n) = a(n+1) - a(n). - Alford Arnold, Jul 19 2000

a(n) = n + floor(n/3) + floor(n/9) + floor(n/27) + ... = n + a(floor(n/3)) = n + A054861(n) = A054861(3n) = (3*n - A053735(n))/2. - Henry Bottomley, May 01 2001

a(n) = Sum_{k>=0} floor(n/3^k). a(n) = Sum_{k=0..floor(log_3(n))} floor(n/3^k), n >= 1. - Hieronymus Fischer, Aug 14 2007

Recurrence: a(n) = n + a(floor(n/3)); a(3n) = 3*n + a(n); a(n*3^m) = 3*n*(3^m-1)/2 + a(n). - Hieronymus Fischer, Aug 14 2007

a(k*3^m) = k*(3^(m+1)-1)/2, 0 <= k < 3, m >= 0. - Hieronymus Fischer, Aug 14 2007

Asymptotic behavior: a(n) = (3/2)*n + O(log(n)), a(n+1) - a(n) = O(log(n)); this follows from the inequalities below. - Hieronymus Fischer, Aug 14 2007

a(n) <= (3n-1)/2; equality holds for powers of 3. - Hieronymus Fischer, Aug 14 2007

a(n) >= (3n-2)/2 - floor(log_3(n)); equality holds for n = 3^m - 1, m > 0. - Hieronymus Fischer, Aug 14 2007

Lim inf (3n/2 - a(n)) = 1/2, for n->oo. - Hieronymus Fischer, Aug 14 2007

Lim sup (3n/2 - log_3(n) - a(n)) = 0, for n->oo. - Hieronymus Fischer, Aug 14 2007

Lim sup (a(n+1) - a(n) - log_3(n)) = 1, for n->oo. - Hieronymus Fischer, Aug 14 2007

G.f.: (Sum_{k>=0} x^(3^k)/(1-x^(3^k)))/(1-x). - Hieronymus Fischer, Aug 14 2007

a(n) = Sum_{k>=0} A030341(n,k)*A003462(k+1). - Philippe Deléham, Oct 21 2011

a(n) ~ 3*n/2 - log(n)/(2*log(3)). - Vaclav Kotesovec, Oct 17 2019

Extensions

Current definition suggested by Jason Earls, Jul 04 2001