A230403 a(n) = the largest k such that (k+1)! divides n; the number of trailing zeros in the factorial base representation of n (A007623(n)).

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1

Offset: 1

Author

Antti Karttunen, Oct 31 2013

Keywords

• nonn
• base
• easy

Comments

Many of the comments given in A055881 apply also here.

From Amiram Eldar, Mar 10 2021: (Start)

The asymptotic density of the occurrences of k is (k+1)/(k+2)!.

The asymptotic mean of this sequence is e - 2 = 0.718281... (A001113 - 2). (End)

Examples

			In factorial number base representation (A007623), the numbers from 1 to 9 are represented as:
n  A007623(n)  a(n) (gives the number of trailing zeros)
1        1       0
2       10       1
3       11       0
4       20       1
5       21       0
6      100       2
7      101       0
8      110       1
9      111       0
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10080
• Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A001113, A055881. Bisection: A230404.

A few sequences related to factorial base representation (A007623): A034968, A084558, A099563, A060130, A227130, A227132, A227148, A227149, A153880.

Analogous sequence for binary system: A007814.

Programs

Mathematica

With[{b = MixedRadix[Range[12, 2, -1]]}, Array[LengthWhile[Reverse@ IntegerDigits[#, b], # == 0 &] &, 105]] (* Michael De Vlieger, Jun 03 2020 *)

Scheme

(define (A230403 n) (if (zero? n) 0 (let loop ((n n) (i 2)) (cond ((not (zero? (modulo n i))) (- i 2)) (else (loop (/ n i) (1+ i)))))))

Formula

a(n) = A055881(n)-1.