A122840 a(n) is the number of 0's at the end of n when n is written in base 10.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0

Offset

1,100

Comments

Greatest k such that 10^k divides n.

a(n) = the number of digits in n - A160093(n).

a(A005117(n)) <= 1. - Reinhard Zumkeller, Mar 30 2010

See A054899 for the partial sums. - Hieronymus Fischer, Jun 08 2012

From Amiram Eldar, Mar 10 2021: (Start)

The asymptotic density of the occurrences of k is 9/10^(k+1).

The asymptotic mean of this sequence is 1/9. (End)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

S. Ikeda and K. Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin., 8 (16) 2013, 63-69.

Formula

a(n) = A160094(n) - 1.

From Hieronymus Fischer, Jun 08 2012: (Start)

With m = floor(log_10(n)), frac(x) = x-floor(x):

a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/10^j))).

a(n) = m + Sum_{j=1..m} (floor(-frac(n/10^j))).

a(n) = A054899(n) - A054899(n-1).

G.f.: g(x) = Sum_{j>0} x^10^j/(1-x^10^j). (End)

a(n) = min(A007814(n), A112765(n)). - Jianing Song, Jul 23 2022

Example

a(160) = 1 because there is 1 zero at the end of 160 when 160 is written in base 10.

Mathematica

a[n_] := IntegerExponent[n, 10]; Array[a, 100] (* Amiram Eldar, Mar 10 2021 *)

Prog

(Haskell)
a122840 n = if n < 10 then 0 ^ n else 0 ^ d * (a122840 n' + 1)
            where (n', d) = divMod n 10
-- Reinhard Zumkeller, Mar 09 2013
(PARI) a(n)=valuation(n, 10) \\ Charles R Greathouse IV, Feb 26 2014
(Python)
def a(n): return len(str(n)) - len(str(int(str(n)[::-1]))) # Indranil Ghosh, Jun 09 2017
(Python)
def A122840(n): return len(s:=str(n))-len(s.rstrip('0')) # Chai Wah Wu, Jul 06 2022
(Python)
A122840 = lambda n: sympy.multiplicity(10, n) # M. F. Hasler, Apr 05 2024

Crossrefs

A007814 is the base 2 equivalent of this sequence.

Cf. A160094, A160093, A001511, A070940, A122841, A027868, A054899, A196563, A196564, A004151, A112765.

Sequence in context: A118553 A102448 A102683 * A083919 A063665 A276306

Adjacent sequences: A122837 A122838 A122839 * A122841 A122842 A122843

Keyword

nonn,base,easy,changed

Author

Reinhard Zumkeller, Sep 13 2006

Status

approved