A112765 Exponent of highest power of 5 dividing n. Or, 5-adic valuation of n.

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

Offset

1,25

Comments

A027868 gives partial sums.

This is also the 5-adic valuation of Fibonacci(n). See Lengyel link. - Michel Marcus, May 06 2017

Links

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

Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.

T. Lengyel, The order of the Fibonacci and Lucas numbers, Fibonacci Quart. 33 (1995), no. 3, 234-239. See Lemma 1 p. 235.

Formula

Totally additive with a(p) = 1 if p = 5, 0 otherwise.

From Hieronymus Fischer, Jun 08 2012: (Start)

With m = floor(log_5(n)), frac(x) = x-floor(x):

a(n) = Sum_{j=1..m} (1 - ceiling(frac(n/5^j))).

a(n) = m + Sum_{j=1..m} (floor(-frac(n/5^j))).

a(n) = A027868(n) - A027868(n-1).

G.f.: Sum_{j>0} x^5^j/(1-x^5^j). (End)

a(5n) = A055457(n). - R. J. Mathar, Jul 17 2012

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/4. - Amiram Eldar, Feb 14 2021

a(n) = 5*Sum_{j=1..floor(log(n)/log(5))} frac(binomial(n, 5^j)*5^(j-1)/n). - Dario T. de Castro, Jul 10 2022

Maple

A112765 := proc(n)
    padic[ordp](n, 5) ;
end proc: # R. J. Mathar, Jul 12 2016

Mathematica

a[n_] := IntegerExponent[n, 5]; Array[a, 105] (* Jean-François Alcover, Jan 25 2018 *)

Prog

(Haskell)
a112765 n = fives n 0 where
   fives n e | r > 0     = e
             | otherwise = fives n' (e + 1) where (n', r) = divMod n 5
-- Reinhard Zumkeller, Apr 08 2011
(PARI) A112765(n)=valuation(n, 5); /* Joerg Arndt, Apr 08 2011 */
(Python)
def a(n):
    k = 0
    while n > 0 and n%5 == 0: n //= 5; k += 1
    return k
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 06 2021

Crossrefs

Cf. A007814, A007949, A112762, A022337, A122840, A027868, A054899, A122841, A160093, A160094, A196563, A196564.

Cf. A343251.

Sequence in context: A073345 A216511 A138088 * A105966 A318950 A319000

Adjacent sequences: A112762 A112763 A112764 * A112766 A112767 A112768

Keyword

nonn,easy

Author

Reinhard Zumkeller, Sep 18 2005

Status

approved